ch04 - part 2 (Cont unifm dist).ppt explained
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DirectionsUse what you have learned in this course to answer th.docx
Directions:
Use what you have learned in this course to answer the following questions. Justify your responses completely. Each question is worth 5 points.
1.
Solve for
n
:
–6(
n
– 8) = 4(12 – 5
n
) + 14
n
.
2
. For
f(
x
) = 2|
x
+3| – 5
, name the type of function and describe each of the three transformations from the parent function
f(
x
) = |
x
|
.
3.
Determine whether
f
(
x
) = –5 – 10
x
+ 6
has a maximum or a minimum value. Find that value and explain how you know.
4.
The median weekly earnings for American workers in 1990 was $412 and in 1999 it was $549. Calculate the average rate of change between 1990 and 1999.
5.
Find the roots of the parabola given by the following equation.
2
x
2+ 5
x
- 9 = 2
x
6.
Describe the end behavior and determine whether the graph represents an odd-degree or an even-degree polynomial function. Then state the number of real zeros.
7.
GEOMETRY
Recall the formula for finding the area of a rectangle. Define a variable for the width and set up an equation to find the dimensions of a rectangle that has an area 144 square inches, given that the length is 10 inches longer than its width.
DIMENSIONS:
Length: Width:
8.
The amount
f
(
t
) of a certain medicine, in milligrams, in a patient’s bloodstream
t
minutes after being taken is given by
f
(
t
) =
.
Find the amount of medicine in the blood after 20 minutes.
9.
Graph
f(
x
) =
x
2 + 2
x
- 3
, label the function’s x-intercepts,
y
-intercept and vertex with their coordinates. Also draw in and label the axis of symmetry.
Image result for x y axis
10.
Determine whether the relation shown is a function. Explain how you know.
73-1.jpg
11.
Solve the inequality and graph the solution on a number line.
–3(5
y
– 4) ≥ 17
12.
Assume that the wooden triangle shown is a right triangle.
a. Write an equation using the Pythagorean Theorem and the measurements provided in the diagram.
Hint: (leg 1)2 + (leg 2)2 = (hypotenuse)2
b. Transform each side of the equation to determine if it is an identity.
13.
Use long division or synthetic division to find the quotient of .
14.
Simplify
(9 + 8 – 6)(4 – 5)
.
15.
Find the inverse of
h(
x
) = .
16.
If
f(
x
) = 2
x
– 1
and
g(
x
) = – 2
, find
[g
◦ f](
x
).
17.
Graph the function
y
= – 2
. Then state the domain and range of the function.
Domain:
Range:
18.
If
f(
x
) = 3
x
2 – 2
and
g(
x
) = 4x + 2
, what is the value of
f
+ g 2
?
The price of a sweatshirt at a local shop is twice the price of a pair of shorts. The price of a T-shirt at the shop is $4 less than the price of a pair of shorts. Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total cost of $136.
19.
Let
w
represent the price of one sweatshirt,
t
represent the price of one T-shirt, and
h
represent the price of one pair of shorts. Write a system of three equations that represents the prices of the clothing.
20.
Solve the system. Find the cost of eac.
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- The mean, variance and standard deviation of a binomial distribution are np, npq, and √npq respectively.
- Examples demonstrate calculating probabilities of outcomes for binomial experiments and determining if results are significantly low or high using the range rule of μ ± 2σ.
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1. Binomial distribution - For experiments with a fixed number of trials, two possible outcomes, and constant probability of success. The probability of x successes is given by the binomial formula.
2. Geometric distribution - For experiments repeated until the first success. The probability of the first success on the xth trial is p(1-p)^(x-1).
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3. An example is provided to demonstrate calculating these values from data on the number of vehicles owned by households.
4. It also introduces continuous random variables and density functions, noting that the probability of any single value is zero due to the infinite number of possible outcomes. The area under the density function curve represents probabilities.
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- All normal distributions can be converted to the standard normal distribution (with μ=0 and σ=1) by subtracting the mean and dividing by the standard deviation.
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- The uniform, normal, and exponential distributions are introduced as examples of continuous distributions. Key properties like expected value and variance are discussed.
- The standard normal distribution is especially important, and its probabilities are provided in tables. Examples show how to calculate probabilities for normal distributions using the tables.
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The document discusses probability distributions, which model the probabilities of outcomes in random phenomena. It covers:
- Discrete and continuous probability distributions, which model outcomes that are discrete or continuous.
- Key properties of distributions like the mean, variance, and standard deviation, which describe the central tendency and variation of outcomes.
- Specific discrete distributions like the binomial and Poisson, which model counts of successes/failures and rare events.
- The normal/Gaussian distribution as the most common continuous distribution, fully described by its mean and variance.
- Standardizing normal variables to the standard normal distribution with mean 0 and variance 1 for easier probability calculations.
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This document discusses the normal distribution and its key properties. It also discusses sampling distributions and the central limit theorem. Some key points:
- The normal distribution is bell-shaped and symmetric. It is defined by its mean and standard deviation. Approximately 68% of values fall within 1 standard deviation of the mean.
- Sample statistics like the sample mean follow sampling distributions. When samples are large and random, the sampling distributions are often normally distributed according to the central limit theorem.
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DirectionsUse what you have learned in this course to answer th.docx
Directions:
Use what you have learned in this course to answer the following questions. Justify your responses completely. Each question is worth 5 points.
1.
Solve for
n
:
–6(
n
– 8) = 4(12 – 5
n
) + 14
n
.
2
. For
f(
x
) = 2|
x
+3| – 5
, name the type of function and describe each of the three transformations from the parent function
f(
x
) = |
x
|
.
3.
Determine whether
f
(
x
) = –5 – 10
x
+ 6
has a maximum or a minimum value. Find that value and explain how you know.
4.
The median weekly earnings for American workers in 1990 was $412 and in 1999 it was $549. Calculate the average rate of change between 1990 and 1999.
5.
Find the roots of the parabola given by the following equation.
2
x
2+ 5
x
- 9 = 2
x
6.
Describe the end behavior and determine whether the graph represents an odd-degree or an even-degree polynomial function. Then state the number of real zeros.
7.
GEOMETRY
Recall the formula for finding the area of a rectangle. Define a variable for the width and set up an equation to find the dimensions of a rectangle that has an area 144 square inches, given that the length is 10 inches longer than its width.
DIMENSIONS:
Length: Width:
8.
The amount
f
(
t
) of a certain medicine, in milligrams, in a patient’s bloodstream
t
minutes after being taken is given by
f
(
t
) =
.
Find the amount of medicine in the blood after 20 minutes.
9.
Graph
f(
x
) =
x
2 + 2
x
- 3
, label the function’s x-intercepts,
y
-intercept and vertex with their coordinates. Also draw in and label the axis of symmetry.
Image result for x y axis
10.
Determine whether the relation shown is a function. Explain how you know.
73-1.jpg
11.
Solve the inequality and graph the solution on a number line.
–3(5
y
– 4) ≥ 17
12.
Assume that the wooden triangle shown is a right triangle.
a. Write an equation using the Pythagorean Theorem and the measurements provided in the diagram.
Hint: (leg 1)2 + (leg 2)2 = (hypotenuse)2
b. Transform each side of the equation to determine if it is an identity.
13.
Use long division or synthetic division to find the quotient of .
14.
Simplify
(9 + 8 – 6)(4 – 5)
.
15.
Find the inverse of
h(
x
) = .
16.
If
f(
x
) = 2
x
– 1
and
g(
x
) = – 2
, find
[g
◦ f](
x
).
17.
Graph the function
y
= – 2
. Then state the domain and range of the function.
Domain:
Range:
18.
If
f(
x
) = 3
x
2 – 2
and
g(
x
) = 4x + 2
, what is the value of
f
+ g 2
?
The price of a sweatshirt at a local shop is twice the price of a pair of shorts. The price of a T-shirt at the shop is $4 less than the price of a pair of shorts. Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total cost of $136.
19.
Let
w
represent the price of one sweatshirt,
t
represent the price of one T-shirt, and
h
represent the price of one pair of shorts. Write a system of three equations that represents the prices of the clothing.
20.
Solve the system. Find the cost of eac.
Normal as Approximation to Binomial
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.6: Normal as Approximation to Binomial
Matlab polynimials and curve fitting
This document discusses polynomial functions in MATLAB. It covers:
- Defining polynomials as coefficient vectors and finding roots.
- Adding, subtracting, multiplying and dividing polynomials using functions like conv and deconv.
- Evaluating and differentiating polynomials with polyval and polyder.
- Using polyfit for polynomial curve fitting to minimize squared errors between a polynomial and data set.
- An example of fitting increasing degree polynomials from 2 to 8 to cosine wave data, showing better fitting with higher degrees.
Binomial probability distributions
The document provides information about binomial probability distributions including:
- Binomial experiments have a fixed number (n) of independent trials with two possible outcomes and a constant probability (p) of success.
- The binomial probability distribution gives the probability of getting exactly x successes in n trials. It is calculated using the binomial coefficient and p and q=1-p.
- The mean, variance and standard deviation of a binomial distribution are np, npq, and √npq respectively.
- Examples demonstrate calculating probabilities of outcomes for binomial experiments and determining if results are significantly low or high using the range rule of μ ± 2σ.
Discrete Probability Distributions
This document discusses several discrete probability distributions:
1. Binomial distribution - For experiments with a fixed number of trials, two possible outcomes, and constant probability of success. The probability of x successes is given by the binomial formula.
2. Geometric distribution - For experiments repeated until the first success. The probability of the first success on the xth trial is p(1-p)^(x-1).
3. Poisson distribution - For counting the number of rare, independent events occurring in an interval. The probability of x events is (e^-μ μ^x)/x!, where μ is the mean number of events.
Similar to ch04 - part 2 (Cont unifm dist).ppt explained (20)
DirectionsUse what you have learned in this course to answer th.docx
DirectionsUse what you have learned in this course to answer th.docx
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- 1. Continuous Probability Distribution * Uniform Distribution * Normal Distribution
- 2. Definition 4-5 Continuous Uniform Distribution - Discrete uniform distribution, f(x) = 1/n
- 3. Figure 4-8 Continuous uniform probability density function. 4-5 Continuous Uniform Distribution Discrete uni. dis
- 4. Mean and Variance 4-5 Continuous Uniform Distribution
- 5. Example 4-5 Continuous Uniform Distribution To know that it is unifm. dist f(x)=1/ (b-a) =1/ (20-0) =0.05
- 6. Figure 4-9 Probability for Example 4-9. 4-5 Continuous Uniform Distribution
- 7. 4-5 Continuous Uniform Distribution
- 8. Text Book Problem • 441 a. F(X) = (x-0.9)/0.15 b. P(X>1.02) = 0.2 c. Here you have to identify the value of x such that the function will equate to 0.9 or 90% i.e., P(X>x)=0.9 x=0.915 d. E(X)=0.976; V(X)=0.00188
- 9. Text Book Problem • 446 (Please note that there is a mistake in the problem. Change 746 with 374 and 752 with 380) a. E(X) = 377; s(X)=1.732 b. P(X<375) = 0.1667 c. P(X>x)=0.95; x=374.3 d. Mean extra cost = 0.004/ container (Hint: mean extra cost = cost* mean extra volume)
- 11. • ‘Bell Shaped’ • Symmetrical • Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: + to X f(X) μ σ 4-6 Normal Distribution
- 13. 4-6 Normal Distribution Figure 4-10 Normal probability density functions for selected values of the parameters and s2.
- 14. f(X) X μ μ+1σ μ-1σ σ σ 68.26% x μ 2σ 2σ x μ 3σ 3σ 95.44% 99.73%
- 15. 4-6 Normal Distribution Some useful results concerning the normal distribution
- 16. 4-6 Normal Distribution Definition : Standard Normal z = (x-)/s
- 17. 4-6 Normal Distribution Example 4-11 Figure 4-13 Standard normal probability density function. Normal distribution table is in Page 718 and Page 719 P(Z<=1.5)=0.93319 P(Z<=1.53)=0.93699
- 20. Text-book problem 4.49 a. P(Z<1.32) =0.9066 c. P(Z>1.45) = 1-P(Z<1.45) (Because the table gives the lower value of Z i.e., P(Z<z) = 0.0735 d. P(Z>-2.15) = 1- 0.0158=0.9842 e. P(-2.34<Z<1.76) ---- see in the next slide 1.45 1 = 100% Non-shaded area = 1-shaded area
- 21. f(X) X 1.76 -2.34 a = P(Z<1.76) P(-2.34<Z<1.76) b = P(Z<-2.34) P(-2.34<Z<1.76) = P(Z<1.76)-P(Z<-2.34) = 0.9511
- 22. Thanks a lot for listening. Remaining portion we will discuss in the next class