Let X1,X2, . . . be a sequence of independent identically distribute.pdf
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Let X1,X2, . . . be a sequence of independent identically distributed random variables with finite mean and variance. Show that the sequence Yn = Xn/n converges to zero, with probability 1. Solution Y1 = X1/1, Y2 = X2/2, Y3 = X3/3... Yn = Xn/n According to Levy\'s theorem, The independent random variables Sn = X1 ,.
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Linear Algebra, please show work, and explanation, thank you!Sol.pdf
Linear Algebra, please show work, and explanation, thank you!
Solution
If b1 belongs to R(T) then T(b1) should be in R(T)
T(b1) = Ab1 =
8
3
5
Find T(b2)=
12
-1
13
T(b1)+T(b2) =
20
2
18
T(b1+b2) = T(7, 5, 1)=
20
2
18
= T(R1)+T(R2).
Linear Algebra Project Urban Population Dynamics This project is.pdf
Linear Algebra Project : Urban Population Dynamics
This project is about population modeling and how linear algebra tools may be used to study it.
Background
Population modeling is useful from dierent perpectives :
1. planners at the city, state, and national level who look at human populations and need
forecasts of
populations in order to do planning for future needs. These future needs include housing,
schools, care
for elderly, jobs, and utilities such as electricity, water and transportation.
2. businesses do population planning so as to predict how the portions of the population that use
their
product will be changing.
3. ecologists use population models to study ecological systems, especially those where
endangered species
are involved so as to try to nd measures that will restore the population.
4. medical researchers treat microorganisms and viruses as populations and seek to understand
the dy-
namics of their populations ; especially why some thrive in certain environments but don\'t in
others.
In human situations, it is normal to take the intervals of 10 years as the census is taken every 10
years.
Thus the age groups would be 0-9, 10-19, 20-29 etc, so 8 or 9 age categories would probably be
appropriate.
The survival fractions would then show the fraction of \"newborns\" (0-9) who survive to age
10, the fraction
of 10 to 19 years old who survive to 20 etc. This type of data is compiled, for example, by
actuaries working
for insurance companies for life and medical insurance purposes.
The basic equations we begin with are
x(k + 1) = Ax(k) k = 0; 1; 2; ::: and x(0) given (0.1)
with solution found iteratively to be
x(k) = Akx(0) (0.2)
Your project
Suppose we are studying the population dynamics of Los Angeles for the purpose of making
planning proposal.
As above, we take the unit of time to be 10 years, and take 7 age groups : 0-9, 10-19,..., 50-59,
60+. Suppose
further that the population distribution as of 1990 is
(3:1; 2:8; 2:0; 2:5; 2:0; 1:8; 2:9)(105)
1
and that the Leslie matrix, A, for this model appears as
A :=
2
666666664
:2 1:2 1:1 :9 :1 0 0
:7 0 0 0 0 0 0
0 :82 0 0 0 0 0
0 0 :97 0 0 0 0
0 0 0 :97 0 0 0
0 0 0 0 :90 0 0
0 0 0 0 0 :87 0
3
777777775
Part One :
Interpret carefully each of the nonzero terms in the matrix. In addition, indicate what factors you
think
might change those numbers (they might be social, economical, political or environmental).
Part Two :
Predict :
what the population distribution will look like in 2000, 2010, 2020, and 2030 ?
what the total population will be in each of these years ?
by what fraction the total population changed each year ?
Additionally, what does your software tell you the largest positive eigenvalue of A is ?
Part Three :
Decide if you believe the population is going to zero, becoming stable, or is unstable in the long
run. Be
sure and describe in your write up how you arrived at your conclusion. If you have decided it is
unstable,
simulate it long enough that the column matrices for.
Linear Algebra please show all work Linear Algebra please show all w.pdf
Linear Algebra please show all work Linear Algebra please show all work Question 1 (5
points). Use mathematical induction to show that 1^2 + 2^2 + ? + n^2 = n(n + 1)(2n + 1)/6, for
all n epsilon N.
Solution
Basis for P(1):
LHS: 12 = 1
RHS: [1(1+1)(2(1)+1)]/6 = (2)(3) / 6 = 1
Basis for P(1) holds
Induction:
Assume P(k) holds for some integer k?1:
12 + 22 + 32 + ... + k2 + = [k(k+1)(2k+1)]/6
Goal is to establish that:
12 + 22 + 32 + ... + k2 + (k+1)2 = [(k+1)(k+2)(2(k+1)+1)]/6
= k(k+1)(2k+1) + (k+1)2 /6
= k(k+1)(2k+1) + 6(k+1)2 / 6
= (k+1)[(k(2k+1) + 6(k+1))] / 6
= (k+1)[2k2+k + 6k+6] / 6
= (k+1)[(2k2+7k+6))] / 6
= (k+1)[(2k2+4k + 3k+6))] / 6
= (k+1)[(2k(k+2) +3(k+2))] / 6
= (k+1)(k+2)(2k+3) / 6
= (k+1)(k+2)(2(k+1)+1) / 6
hence proof :).
Linear Algebra Help- I cant understand where to start in this. Could.pdf
Linear Algebra Help- I cant understand where to start in this. Could someone explain: Find the
matrix that projects any vector onto the 45 degree line, through (0, 0), (1, 1). Call it P45. : The
1st column is What is its nullspace? What is its column space ?
Solution
2a)...HOPE YOU MEAN R2.
limit as x goes to infinity of cos(1 x) Solutionrepresent .pdf
limit as x goes to infinity of: cos(1 / x)
Solution
represent cos(1/x) in series form :
cos x = 1 - x2 /2! + x4 /4! ..........
cos(1/x) = 1 - (1/2!)(1/x2 ) + (1/4!)(1/x4 ) ...........
(1 - 1/2! (1/x2 ) + (1/4!)(1/x4 ).................) x
this is form (-> 1)infinity
e ( -1/2! 1/(x*x))x = e 0 = 1 answer !!!!.
Lily and sara each had an equal amount of money at first. After Lily.pdf
Lily and sara each had an equal amount of money at first. After Lily spent 18 and sara spent 25
dollars, Lily had twice as much as as sara.How much money did each of them have at first?
Solution
Let Lily and Sara each had x dollars THen, x-18=2(x-25) =>x-18=2x-50 =>-x=-
32 =>x=32 ans.
Likely reasons for inaccurate control limits would include which of .pdf
Likely reasons for inaccurate control limits would include which of the following?
The engineering parameter for variance is unknown.
The engineers were underpaid for their work.
There was insufficient preliminary sampling.
Process variation was not zero, as expected.
Solution
The engineers were underpaid for their work..
lim 4x^3+6x^2-2 2x^3 -4x+5 when x approaches negative infinity .pdf
lim 4x^3+6x^2-2/ 2x^3 -4x+5 when x approaches negative infinity
Solution
divide by the highest power of x both in numerator and denomiantor
[4 + 6/x - 2/x3 ]/[2 - 4/x2 + 5/x3]
now put x -> - infinity
4/2 = 2
answer !!!.
Recommended
Linear Algebra, please show work, and explanation, thank you!Sol.pdf
Linear Algebra, please show work, and explanation, thank you!
Solution
If b1 belongs to R(T) then T(b1) should be in R(T)
T(b1) = Ab1 =
8
3
5
Find T(b2)=
12
-1
13
T(b1)+T(b2) =
20
2
18
T(b1+b2) = T(7, 5, 1)=
20
2
18
= T(R1)+T(R2).
Linear Algebra Project Urban Population Dynamics This project is.pdf
Linear Algebra Project : Urban Population Dynamics
This project is about population modeling and how linear algebra tools may be used to study it.
Background
Population modeling is useful from dierent perpectives :
1. planners at the city, state, and national level who look at human populations and need
forecasts of
populations in order to do planning for future needs. These future needs include housing,
schools, care
for elderly, jobs, and utilities such as electricity, water and transportation.
2. businesses do population planning so as to predict how the portions of the population that use
their
product will be changing.
3. ecologists use population models to study ecological systems, especially those where
endangered species
are involved so as to try to nd measures that will restore the population.
4. medical researchers treat microorganisms and viruses as populations and seek to understand
the dy-
namics of their populations ; especially why some thrive in certain environments but don\'t in
others.
In human situations, it is normal to take the intervals of 10 years as the census is taken every 10
years.
Thus the age groups would be 0-9, 10-19, 20-29 etc, so 8 or 9 age categories would probably be
appropriate.
The survival fractions would then show the fraction of \"newborns\" (0-9) who survive to age
10, the fraction
of 10 to 19 years old who survive to 20 etc. This type of data is compiled, for example, by
actuaries working
for insurance companies for life and medical insurance purposes.
The basic equations we begin with are
x(k + 1) = Ax(k) k = 0; 1; 2; ::: and x(0) given (0.1)
with solution found iteratively to be
x(k) = Akx(0) (0.2)
Your project
Suppose we are studying the population dynamics of Los Angeles for the purpose of making
planning proposal.
As above, we take the unit of time to be 10 years, and take 7 age groups : 0-9, 10-19,..., 50-59,
60+. Suppose
further that the population distribution as of 1990 is
(3:1; 2:8; 2:0; 2:5; 2:0; 1:8; 2:9)(105)
1
and that the Leslie matrix, A, for this model appears as
A :=
2
666666664
:2 1:2 1:1 :9 :1 0 0
:7 0 0 0 0 0 0
0 :82 0 0 0 0 0
0 0 :97 0 0 0 0
0 0 0 :97 0 0 0
0 0 0 0 :90 0 0
0 0 0 0 0 :87 0
3
777777775
Part One :
Interpret carefully each of the nonzero terms in the matrix. In addition, indicate what factors you
think
might change those numbers (they might be social, economical, political or environmental).
Part Two :
Predict :
what the population distribution will look like in 2000, 2010, 2020, and 2030 ?
what the total population will be in each of these years ?
by what fraction the total population changed each year ?
Additionally, what does your software tell you the largest positive eigenvalue of A is ?
Part Three :
Decide if you believe the population is going to zero, becoming stable, or is unstable in the long
run. Be
sure and describe in your write up how you arrived at your conclusion. If you have decided it is
unstable,
simulate it long enough that the column matrices for.
Linear Algebra please show all work Linear Algebra please show all w.pdf
Linear Algebra please show all work Linear Algebra please show all work Question 1 (5
points). Use mathematical induction to show that 1^2 + 2^2 + ? + n^2 = n(n + 1)(2n + 1)/6, for
all n epsilon N.
Solution
Basis for P(1):
LHS: 12 = 1
RHS: [1(1+1)(2(1)+1)]/6 = (2)(3) / 6 = 1
Basis for P(1) holds
Induction:
Assume P(k) holds for some integer k?1:
12 + 22 + 32 + ... + k2 + = [k(k+1)(2k+1)]/6
Goal is to establish that:
12 + 22 + 32 + ... + k2 + (k+1)2 = [(k+1)(k+2)(2(k+1)+1)]/6
= k(k+1)(2k+1) + (k+1)2 /6
= k(k+1)(2k+1) + 6(k+1)2 / 6
= (k+1)[(k(2k+1) + 6(k+1))] / 6
= (k+1)[2k2+k + 6k+6] / 6
= (k+1)[(2k2+7k+6))] / 6
= (k+1)[(2k2+4k + 3k+6))] / 6
= (k+1)[(2k(k+2) +3(k+2))] / 6
= (k+1)(k+2)(2k+3) / 6
= (k+1)(k+2)(2(k+1)+1) / 6
hence proof :).
Linear Algebra Help- I cant understand where to start in this. Could.pdf
Linear Algebra Help- I cant understand where to start in this. Could someone explain: Find the
matrix that projects any vector onto the 45 degree line, through (0, 0), (1, 1). Call it P45. : The
1st column is What is its nullspace? What is its column space ?
Solution
2a)...HOPE YOU MEAN R2.
limit as x goes to infinity of cos(1 x) Solutionrepresent .pdf
limit as x goes to infinity of: cos(1 / x)
Solution
represent cos(1/x) in series form :
cos x = 1 - x2 /2! + x4 /4! ..........
cos(1/x) = 1 - (1/2!)(1/x2 ) + (1/4!)(1/x4 ) ...........
(1 - 1/2! (1/x2 ) + (1/4!)(1/x4 ).................) x
this is form (-> 1)infinity
e ( -1/2! 1/(x*x))x = e 0 = 1 answer !!!!.
Lily and sara each had an equal amount of money at first. After Lily.pdf
Lily and sara each had an equal amount of money at first. After Lily spent 18 and sara spent 25
dollars, Lily had twice as much as as sara.How much money did each of them have at first?
Solution
Let Lily and Sara each had x dollars THen, x-18=2(x-25) =>x-18=2x-50 =>-x=-
32 =>x=32 ans.
Likely reasons for inaccurate control limits would include which of .pdf
Likely reasons for inaccurate control limits would include which of the following?
The engineering parameter for variance is unknown.
The engineers were underpaid for their work.
There was insufficient preliminary sampling.
Process variation was not zero, as expected.
Solution
The engineers were underpaid for their work..
lim 4x^3+6x^2-2 2x^3 -4x+5 when x approaches negative infinity .pdf
lim 4x^3+6x^2-2/ 2x^3 -4x+5 when x approaches negative infinity
Solution
divide by the highest power of x both in numerator and denomiantor
[4 + 6/x - 2/x3 ]/[2 - 4/x2 + 5/x3]
now put x -> - infinity
4/2 = 2
answer !!!.
like many entreneurs, Tony Smith started Sonic as a sole proprietors.pdf
like many entreneurs, Tony Smith started Sonic as a sole proprietorship, then took on a partner,
and eventually offered franchises. what advantages did he enjoy at each stage of Sonic\'s
development? What disadvantages did he face?
Solution
There are a number of disadvantages of the company organization as a sole proprietorship:
Advantages sole proprietorship:
Disadvantages of changing the company organization to partnership:
Advantages partnership:
Advantages of changing the company organization to franchises:
Disadvantages of changing the company organization to franchises:.
Like a sole proprietorship statement of owners equity, the stateme.pdf
Like a sole proprietorship statement of owner\'s equity, the statement of partners\' equity will
show all the partners\' capital accounts as one account.
T/F?
Solution
False,
The Partner\'s equity will show all the partner\'s capital account in separate account because it
has to separate all the individual transactions of each partner..
Lewotontin wrote a critical review of The Social Organization of Sex.pdf
Lewotontin wrote a critical review of The Social Organization of Sexuality by E.O. Laumann
and others. Laumann was using data from a sample survey, in which respondents answer
questions about their sexual behavior, including the number of partners in the previous five-year
period. On average, among heterosexuals, men reported having about twice as many partners as
women. Lewotontin thought this was a serious inconsistency, showing that respondants \"are
telling themsevles and other enormous lies\" Laumann replied that you should not use averages
to summarize such skewed and long-tailed distributions.
a) Why is it inconsistent for men to report having twice as many partners as women?
Solution
Yes the data is inconsistent, for men to report having twice as many partners as women,
because of the survey data is response variable.
because of the survey data is response variable..
Let Z_{n} denote the sample mean for a random sample of size n from.pdf
Let Z_{n} denote the sample mean for a random sample of size n from
the standard normal distribution. For n = 1, 4 and 16
1. sketch the probability.distibution.function of each Z_{n} in the same coordinate system,
2. compute the quartiles of each Z_{n}
Solution
We begin our journey into inferential statistics. Most of the time the population
mean and population standard deviation are impossible or too expensive to determine exactly.
Two of the major tasks of a statistician is to get an approximation to the mean and analyze how
accurate the approximation is. The most common way of accomplishing this task is by using
sampling techniques. Out of the entire population the researcher obtains a (hopefully random)
sample from the population and uses the sample to make inferences about the population. From
the sample the statistician computes several numbers such as the sample size, the sample mean,
and the sample standard deviation. The numbers that are computed from the sample are called
statistics..
LetX_1, X_2, . . . , X_nbe i.i.d. from distributionF 1. IfFis t.pdf
LetX_1, X_2, . . . , X_nbe i.i.d. from distributionF:
1. IfFis theUniform(-theta,0) distribution, find the Maximum likelihood explanation oftheta.
Show work for credit.
2. IfFis theUniform(?theta,theta) distribution, find the Maximum likelihood estimation oftheta.
Show work for credit.
Solution
Suppose there is a samplex1,x2, ...,xnofnindependent and identically distributedobservations,
coming from a distribution with an unknownprobability density functionf0(.
Lets say you frequently drink wine thats been rated highly by Ro.pdf
Let\'s say you frequently drink wine that\'s been rated highly by Robert Parker, one of the wine
experts. His ratings were demonstrated to be reliable and you find that you usually agree with
Robert Parker. How does this observation reflect the concept of validity?
Solution
As you know that validity is \"the degree to which evidence and theory support the
interpretations of test scores\".
Since the rating of him is reliable to as as you agree his experties so the Concept of validity is
True here..
Let X1,X2,X3 and X4 be independent random variables. Assume that X2.pdf
Let X1,X2,X3 and X4 be independent random variables. Assume that X2,X3 and X4 each are
poison distributed with mean 5, suppose Y=X1+X2+X3+X4~ POI(25). What is the distribution
of X1.
Solution
pendent random variables. Assume that X2,X3 and X4 each are poison distributed with mean 5,
suppose Y=X1+X2+X3+X4~ POI(25). What is the distribution of X1.
Y will be poissone with mean of 5 + 5 + 5 + X1 so
5 + 5 +5 + x1 = 25 ( but given is 25)
x1 = 25 - 15
x1 = 10
X1 is poison random with mean = 10.
Let X1,X2,X3....X25 be a random sample from a population with mean o.pdf
Let X1,X2,X3....X25 be a random sample from a population with mean of 100 and standard
deviation of 12. Then the standard deviation of the sample mean Xbar, referred to as the standard
error, is _________. The standard error decreases when the sample size________.
Solution
S.E(Xbar)= sd/sqrt (N) *sqrt ((N-n)/(N-1)) S.E=12/sqrt(33) increases.
Let Y1 and Y2 be independent and uniformly distributed over the inte.pdf
Let Y1 and Y2 be independent and uniformly distributed over the interval (0,1). Find the p(2Y1
Solution
FOLLOW THIS a) Probability density of U is also uniform, but it is a square It can
be written as Pr(U2) = 1 for 0<=Y1<=1,0<=Y2<=1 b) E(U2) = 0.5*0.5 = 0.25 V(U2) = E^2(U2)
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Let Y1 be the number of success in n = 10 independent Bernoulli trials with probability of
success θ. Suppose the four possible values for θ are Ω = {0.2, 0.4, 0.6, 0.8}. We observe Y1 =
7.
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non-informative prior).
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0.2, P(θ = 0.6) = 0.3, P(θ = 0.8) = 0.4.
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function.
Solution
plz repost the question.
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Y1, Y2 , Y3 ,Y4 Let them be in increasing order then median is the average of
middle values so it must be in between y2 and y3 so we have two values greater than median
so the probability that the largest exceeds the median = 2/4 = 1/2 or 0.5.
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Hi, There are lot of ? symbols in the question. Can you please re-post the question.
Thanks..
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There are 5! = 120 possible ways to order 5 objects. Each has the same probability
since they are independent rv\'s. So the answer is 1/120..
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The moment generating function is (p1e^t1` + p2e^t2 + 1 - p1 - p2)^n
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Then, var(x1,x2) = E(x1x2) - E(x1)E(x2)
We know that E(x1) = np1 and E(x2) = np2. We can also see this from the first derivative of the
moment generating function.
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(Note that this gives us E(x1) = np1; calculating for x2 similiarly gives us E(x2) = np2)
Then, the derivative with respect to t2 is
np1e^t1(n-1)p2e^t2(p1e^t1` + p2e^t2 + 1 - p1 - p2)^(n-2)
Evaluating this at t1 = t2 = ...=tk-1 = 0, we getnp1(n-1)p2(1)^n-2=
np1(n-1)p2
Then the covariance isnp1(n-1)p2 - np1(np2) =
p1p2(n^2-n)-n^2p1p2 =
p1p2(n^2-n-n^2) =
-np1p2.
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like many entreneurs, Tony Smith started Sonic as a sole proprietors.pdf
like many entreneurs, Tony Smith started Sonic as a sole proprietorship, then took on a partner,
and eventually offered franchises. what advantages did he enjoy at each stage of Sonic\'s
development? What disadvantages did he face?
Solution
There are a number of disadvantages of the company organization as a sole proprietorship:
Advantages sole proprietorship:
Disadvantages of changing the company organization to partnership:
Advantages partnership:
Advantages of changing the company organization to franchises:
Disadvantages of changing the company organization to franchises:.
Like a sole proprietorship statement of owners equity, the stateme.pdf
Like a sole proprietorship statement of owner\'s equity, the statement of partners\' equity will
show all the partners\' capital accounts as one account.
T/F?
Solution
False,
The Partner\'s equity will show all the partner\'s capital account in separate account because it
has to separate all the individual transactions of each partner..
Lewotontin wrote a critical review of The Social Organization of Sex.pdf
Lewotontin wrote a critical review of The Social Organization of Sexuality by E.O. Laumann
and others. Laumann was using data from a sample survey, in which respondents answer
questions about their sexual behavior, including the number of partners in the previous five-year
period. On average, among heterosexuals, men reported having about twice as many partners as
women. Lewotontin thought this was a serious inconsistency, showing that respondants \"are
telling themsevles and other enormous lies\" Laumann replied that you should not use averages
to summarize such skewed and long-tailed distributions.
a) Why is it inconsistent for men to report having twice as many partners as women?
Solution
Yes the data is inconsistent, for men to report having twice as many partners as women,
because of the survey data is response variable.
because of the survey data is response variable..
Let Z_{n} denote the sample mean for a random sample of size n from.pdf
Let Z_{n} denote the sample mean for a random sample of size n from
the standard normal distribution. For n = 1, 4 and 16
1. sketch the probability.distibution.function of each Z_{n} in the same coordinate system,
2. compute the quartiles of each Z_{n}
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We begin our journey into inferential statistics. Most of the time the population
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accurate the approximation is. The most common way of accomplishing this task is by using
sampling techniques. Out of the entire population the researcher obtains a (hopefully random)
sample from the population and uses the sample to make inferences about the population. From
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and the sample standard deviation. The numbers that are computed from the sample are called
statistics..
LetX_1, X_2, . . . , X_nbe i.i.d. from distributionF 1. IfFis t.pdf
LetX_1, X_2, . . . , X_nbe i.i.d. from distributionF:
1. IfFis theUniform(-theta,0) distribution, find the Maximum likelihood explanation oftheta.
Show work for credit.
2. IfFis theUniform(?theta,theta) distribution, find the Maximum likelihood estimation oftheta.
Show work for credit.
Solution
Suppose there is a samplex1,x2, ...,xnofnindependent and identically distributedobservations,
coming from a distribution with an unknownprobability density functionf0(.
Lets say you frequently drink wine thats been rated highly by Ro.pdf
Let\'s say you frequently drink wine that\'s been rated highly by Robert Parker, one of the wine
experts. His ratings were demonstrated to be reliable and you find that you usually agree with
Robert Parker. How does this observation reflect the concept of validity?
Solution
As you know that validity is \"the degree to which evidence and theory support the
interpretations of test scores\".
Since the rating of him is reliable to as as you agree his experties so the Concept of validity is
True here..
Let X1,X2,X3 and X4 be independent random variables. Assume that X2.pdf
Let X1,X2,X3 and X4 be independent random variables. Assume that X2,X3 and X4 each are
poison distributed with mean 5, suppose Y=X1+X2+X3+X4~ POI(25). What is the distribution
of X1.
Solution
pendent random variables. Assume that X2,X3 and X4 each are poison distributed with mean 5,
suppose Y=X1+X2+X3+X4~ POI(25). What is the distribution of X1.
Y will be poissone with mean of 5 + 5 + 5 + X1 so
5 + 5 +5 + x1 = 25 ( but given is 25)
x1 = 25 - 15
x1 = 10
X1 is poison random with mean = 10.
Let X1,X2,X3....X25 be a random sample from a population with mean o.pdf
Let X1,X2,X3....X25 be a random sample from a population with mean of 100 and standard
deviation of 12. Then the standard deviation of the sample mean Xbar, referred to as the standard
error, is _________. The standard error decreases when the sample size________.
Solution
S.E(Xbar)= sd/sqrt (N) *sqrt ((N-n)/(N-1)) S.E=12/sqrt(33) increases.
Let Y1 and Y2 be independent and uniformly distributed over the inte.pdf
Let Y1 and Y2 be independent and uniformly distributed over the interval (0,1). Find the p(2Y1
Solution
FOLLOW THIS a) Probability density of U is also uniform, but it is a square It can
be written as Pr(U2) = 1 for 0<=Y1<=1,0<=Y2<=1 b) E(U2) = 0.5*0.5 = 0.25 V(U2) = E^2(U2)
- 0.25^2 = 7/144=0.04861111.
Let Y1 be the number of success in n = 10 independent Bernoulli tria.pdf
Let Y1 be the number of success in n = 10 independent Bernoulli trials with probability of
success θ. Suppose the four possible values for θ are Ω = {0.2, 0.4, 0.6, 0.8}. We observe Y1 =
7.
(a) Find the posterior distribution for θ if we use a uniform prior distribution (i.e. a vague,
non-informative prior).
(b) Find the posterior distribuiton for θ if the prior distribution is P(θ = 0.2) = 0.1, P(θ = 0.4) =
0.2, P(θ = 0.6) = 0.3, P(θ = 0.8) = 0.4.
HINT: You may wish to use R by installing the Bolstad package and utilizing the binodp
function.
Solution
plz repost the question.
Let Y have an exponential distribution with mean . Prove that W.pdf
Let Y have an exponential distribution with mean ?. Prove that W=?y has a weibull density
with ?= ? and m=2
Solution
f(y,ß) = (1/ß)*e^(-y/ß) and F(y,ß) = 1-e^(-y/ß) W = vy = y^(0.5) now put y^0.5 in
place of y in F(y,ß) we get = 1-e^{(-(y)/ß)^0.5} for weibull distribution , F( x,m,a) = 1-e^{[-
(x/a)]^(1/m)} comparing to the above 1/m = 0.5 ; m =2 and a=ß.
Let Y be a continuous random variable and define m as the median of .pdf
Let Y be a continuous random variable and define m as the median of the distribution (not the
sample median). If Y1,..,Y4 are independent random variables from the same continuous
distribution, what is the probability that the largest exceeds the median?
Solution
Y1, Y2 , Y3 ,Y4 Let them be in increasing order then median is the average of
middle values so it must be in between y2 and y3 so we have two values greater than median
so the probability that the largest exceeds the median = 2/4 = 1/2 or 0.5.
Let X1,...,Xn represent a random sample from the discrete distributi.pdf
Let X1,...,Xn represent a random sample from the discrete distribution having the pmf f(x;?)=
(?^x)((1-?)^(1-x)), x=0,1, 0
Solution
Hi, There are lot of ? symbols in the question. Can you please re-post the question.
Thanks..
Let X1,...,X5 be Independent continuous random variables having comm.pdf
Let X1,...,X5 be Independent continuous random variables having common distribution function
\"F\" and density \"f\". Set
I = P(X1X3X5). Evaluate I. Please show work. Thanks. :-)
Solution
There are 5! = 120 possible ways to order 5 objects. Each has the same probability
since they are independent rv\'s. So the answer is 1/120..
Let X1 and X2 have a trinomial distribution. Differentiate the momen.pdf
Let X1 and X2 have a trinomial distribution. Differentiate the moment-generating function to
show that their covariance is -np1p2
Solution
The moment generating function is (p1e^t1` + p2e^t2 + 1 - p1 - p2)^n
Using the derivative with repect to t1 and t2 gives us E[x1x2]
Then, var(x1,x2) = E(x1x2) - E(x1)E(x2)
We know that E(x1) = np1 and E(x2) = np2. We can also see this from the first derivative of the
moment generating function.
The derivative with respect to t1 is np1e^t1(p1e^t1` + p2e^t2 + 1 - p1 - p2)^(n-1)
(Note that this gives us E(x1) = np1; calculating for x2 similiarly gives us E(x2) = np2)
Then, the derivative with respect to t2 is
np1e^t1(n-1)p2e^t2(p1e^t1` + p2e^t2 + 1 - p1 - p2)^(n-2)
Evaluating this at t1 = t2 = ...=tk-1 = 0, we getnp1(n-1)p2(1)^n-2=
np1(n-1)p2
Then the covariance isnp1(n-1)p2 - np1(np2) =
p1p2(n^2-n)-n^2p1p2 =
p1p2(n^2-n-n^2) =
-np1p2.
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