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Engineering Statistics &
Linear Algebra
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Module1-Lec1
Single Random variable
SECAB Institute of Engineering and Technology
Vijayapura
Dr.Noorullah Shariff C 15/23/2020

(Single) Random Variable
• A random variable is a function that assigns a real number, called an
observation, to each outcome in S. It is denoted as
X(a) = xa (2.1)
• The domain of the random variable X is all outcomes, such as a, in S.
• Its range is all observations, such as x , that are in Sx.• Its range is all observations, such as xa, that are in Sx.
• Note:
• A Sample Space S contains all possible outcomes from random experiment.
• The sample space SX is the collection of all real numbers that result from the
outcome of S.
• By convention, random variables are denoted by uppercase letters near the
end of the alphabet: U, V, ... , Z, although exceptions will be made to this
convention from time to time.
5/23/2020 Dr.Noorullah Shariff C 2

Examples of the application of r.v’s
1. The noises n(t) in a communication link are elements of S. The measured
average power of each n(t) is a real number and is an observation in Sx.
2. Manufactured products in use, serving customers, are elements of S. The
measured time to failure of each product is an observation in Sxx
3. Transistors of a particular group, or type, are elements in S. The
measured maximum switching speed at which each transistor can
operate is an observation in Sx.
4. Programs that may be held temporarily in a computer's queue are
elements in S. Counting the number of programs in the queue at a given
time gives observations in Sx.

Cumulative Distribution Functions (cdf)
• cdf for a rv is defined as
• It is conventional to write
• In general, we could have several different cumulative distribution
functions FU(u), FV(v),... , FZ(z) for the different random variables U, V,functions FU(u), FV(v),... , FZ(z) for the different random variables U, V,
... , Z. The argument of a cumulative distribution function is an
independent variable.
• If the independent variable x= ∞ (2 2) gives
• Fx(∞) is the probability that observa ons X(a) are less than or equal to inﬁnity,
which, of course, is a certainty.

• If the independent variable is x = -∞, (2.2) gives
Fx(-∞) is the probability that observa ons X(a) are less than or equal to minus
infinity, which is the impossible event.
• For all other values of the independent variable x,
• We can also see that if a pair of independent variables, x1 and x2, are
chosen such that x2 > x1, then
• We can also see that if a pair of independent variables, x1 and x2, are
chosen such that x2 > x1, then
• That is, a cumulative distribution function defined by (2.2) must be
monotone non-decreasing. Saying this in another way, the derivative,
if it exists, of a cumulative distribution function must always be non-
negative:

Itisconventionaltowrite
Itisconventionaltowrite

Probability Models
• In the process of defining probability models, the probability density
function (pdf) has a major role.
• The pdf is denoted as fx(x) for a random variable X, and is defined as
when the derivative of the cdf exists.
pdfandacdfareinversesofeachother.
when the derivative of the cdf exists.
• In general, we could have several different probability density
functions fu(u).fv(v), ... .fz(z) for the different random variables U, V,
... , Z. The argument of a pdf is an independent variable.
• The inverse of (2.9) is
Apdfandacdfareinversesofeach

Continuous Random Variables
• Here we assume that the cdf is a continuous function and that, except
at a finite number of points, the derivative of Fx(x) in (2.9) exists.
• Some general features of pdf’s and cdf’s are,
• Combining (2.7) and (2.9), we see that a pdf can never be negative for any
value of its independent variable:
∞• When x = -∞, (2.1) gives the cdf a value of zero-a result that we have seen
before in (2.4).
• When x = +∞, (2.10) with (2.3) allows us to write the very important relation
The area under a pdf curve is always 1.

• Using (2.6) and (2. IO), we can write
• Essentially (2.13) says that the area under a pdf curve over some
specific interval in Sx is the probability observations of a random
variable occurring in that interval. A useful approximation of (2.13) is
• Finally, we can see from (2.14) that when Δx=0, the probability that a
random variable exactly equals some specific value is zero:

• For any random variable with a continuous cdf, (2.15) gives us some
flexibility in the use of equalities and inequalities as indicated here.
• Example 2.1: Uniform Distribution
• For uniformly distributed random variable, observations are equally likely to
occur in some interval.
For example, the phase of the sinusoidal carrier in an amplitude modulation• For example, the phase of the sinusoidal carrier in an amplitude modulation
system is arbitrary and may be found to be equally likely between ±π radians.
• Figure 2.3 shows a pdf and a cdf for a random variable Y uniformly distributed
between y1 and y2. The pdf is

• And the cdf for a uniform random variable is, using (2.1O),
• All requirements for a continuous cdf and its associated pdf are met:
• The pdf is always non-negative, and the area under its curve is 1.• The pdf is always non-negative, and the area under its curve is 1.
• The cdf is continuous and non-decreasing from 0 on the left to 1 on the right.
• The derivative of the cdf exists everywhere except at y = y1 and y = y2.

• Example 2.2: Exponential Distribution
• Exponential random variables for example occur in discussions of
failure rates in reliability and in some queuing applications.
• See Figure 2.4 for plots of a typical exponential random variable pdf
and cdf.
• The pdf of an exponential random variable is
• And the exponential cdf is, using (2.10),
• λ > 0, is rate constant

• The pdf is always non-negative, and the area under its curve is 1.
• The derivative of the cdf exists everywhere except at x = 0.

Linear Algebra
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Single Random Variable
Gaussian Distribution
Vijayapura

Example 2.3: Gaussian distribution
• Whenever the observations are measured repeatedly and are
independent then
• the sum of the observations tends to be what we call Gaussian
• Gaussian: distribute like the classical bell-shaped curve.
• For Example: Thermal noise in a resistor has voltage values that
distribute "Gaussian" because
• that noise voltage results from the additive effect of the motion of many
thermally agitated electrons.

• The pdf for the normalized Gaussian random variable (shown in
Figure 2.5(a)) is
• Using (2.10). the associated cdf (shown in Fig 2,5b) is
• The pdf is always non-negative, and the area under its curve is 1.
• The derivative of the cdf exists everywhere.

• The integral (2.22) cannot be evaluated in closed form for arbitrarily,
but is tabulated numerically.
• When this is done, the notation ϕ(z) =FZ(z) is often used.
• Appendix D contains tables of ϕ(z) for 0 ≤ z ≤ 3.00.
• The same table can be used to find values ϕ(z) when z is negative,
-3.00 ≤ z ≤ 0.
• In this case.• In this case.
• Equation (2.24) is valid because the normalized Gaussian pdf is
symmetrical about 0(see Figure 2.5a).
• Then, FZ(-z) = 1 - FZ(z), which is same as (2.24 ).

• Let z= 0.9347, find (z). From the table in Appendix D,
• ϕ(0.93) = 0.8238
• ϕ (0.94) = 0.8264
• Interpolating, ϕ(0.9347) = 0.8250.
• We also note that
• ϕ(-0.9347) = 1-ϕ(0.9347) = 0.1750
• Finding the inverse of ϕ(z) = y, z= ϕ-1(y), may also be done using the table in
Appendix D.
= 0.8238+(0.9347-0.93)/(0.94-0.93)*(0.8264-0.8238)
Appendix D.
• Suppose that we need to find z in ϕ(z)=0.6000.
• From the table,
• ϕ(0.2500) = 0.5987
• ϕ(0.2600) = 0.6026
• Interpolating, we find z = 0.2533:
• ϕ(0.2533) = 0.6000
For 0.6026-0.5987=0.0039  0.26-0.25=0.01 then
For 0.6000-0.5987=0.0013(0.0013/0.0039)*0.01= 0.0033
For 0.6000  z=0.2500+ 0.0033 = 0.2533
=0.2500+ (0.6000-0.5987)/(0.6026-0.5987)*(.26/.25)

AppendixD
AppendixD

ANS=0.4069
ANS=0.1619
ANS=0.3118

z)=1-FZ(z)
FZ(-0.505) = 1 - FZ(0.505),
FZ(-z)=

• =FZ(1.456) - FZ(-0.505)
• =FZ(1.456) –( 1 - FZ(0.505))
• =(0.9265+ (1.456-1.45)/(1.46-1.45)*(0.9279-0.9265))- (1- 0.7088)
• =0.9273-0.2912• =0.9273-0.2912
• =0.6361

• =0.9916+(2.392-2.39)/(2.4-2.39)*(0.9918-0.9916)-(1-(0.8413+(1.003-
1)/(1.01-1)*(0.8438-0.8413)))
• =0.99164-(1-0.84205)
• =0.99164-0.15795• =0.99164-0.15795
• =0.8337

• =0.34 +(0.6340-0.6331)/(0.6368-0.6331)*(0.35-0.34)
• =0.34243

fX(x)=c(x-6)
fY(y)=c(y-3)
fZ(z)=c(z-4)

=FX(0.28) - FX(-0.37)=0.4758

=FX(7.6) - FX(-5.3)=0.362

=FX(4.6) - FX(1.3)=0.0638

Linear Algebra
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Discrete Random Variables, Mixed Random
Variables
Vijayapura

Discrete Random Variables
• A discrete random variable is a variable which can only take a
countable number of values. The variable is said to be random if the
sum of the probabilities is one.
• Example 2.4 :• Example 2.4 :
• Table 2.1 gives an example of
Discrete Random Variable with
Probabilities
• Total Probability =1
=fX(x)

• The cdf for this can be written in terms of unit step functions.
• The pdf can be written in terms of unit impulse functions

• Generalizing for any situation involving a finite number of discrete
random variables,
• When the range of observations SX contains only discrete values, then
X is a Discrete Random Variable.
• A probability Pi associated with a discrete random variable is called a
Probability Mass Function (pmf).Probability Mass Function (pmf).
• When all the discrete observations in SX are considered, their
probabilities must, according to Axiom I, sum to 1:

• For discrete random variables
• The cdf for a discrete random variable may be written as
• then the pdf of Random Variables is

• Example 2.5
• Consider Table
(=fX(x)) (=FX(x))
(cdf)
(pmf or pdf)

• Bernoulli random experiment, produces two mutually exclusive
events A & .
• The probabilities of these event, A & are denoted as
• Using a Bernoulli probability model, a counting random variable X is
assigned the integers 1 and 0 as follows:
• Thus Sx={1,0}
• Now, consider a Binomial trial of order n (i.e., n independent
Bernoulli trial), each with outcome S={ A, } . Then the counting
function is

• Example 2.6
• Table 1.3 illustrates a binomial random
variable of order n=10.
• The first column gives values of k, i.e., the
number of times that a Bernoulli event A can be
counted in a trial.
number of times that a Bernoulli event A can be
counted in a trial.
• The second column gives the probability mass
function for the parameters specified in the
table.
• For example, given the parameters in Table 1.3, the
probability of finding k = 4 events A in a binomial trial
of order n = 10 is P{X = 4} = 0.1460.
• The third column lists the cumulative sum for the
pmf, with the parameters given in·
• Table 1.3. the probability of finding k = 4 or fewer
events A in a binomial trial of order n = 10 is

Mixed random variables
• A random variable that contains features of both a continuous and a
discrete random variable is called a mixed random variable.
• Mixed random variable uses the techniques already developed for the
continuous and discrete random variables.
• Example 2.7
• Suppose a random variable X has the cdf shown in Figure 2.8(a).• Suppose a random variable X has the cdf shown in Figure 2.8(a).
• The cdf illustrated is continuous at all values of x except when x = 2, where
there is a discontinuity of 0.2.
• The slope of the cdf is 1/5 when 0 < x < 2 and 2/5 when 2 < x < 3.
• Therefore, using (2.9) the pdf associated with this cdf is as shown in Figure
2.8(b).
• Thus, the area under the plot in Figure 2.8(b) is

7/307/30

1

FX(x) 4

Linear Algebra
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Expectations
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Expectations
• Expectation of a Random Variable X is written as
• This expansion put emphasis on Expectation Operator E[.].• This expansion put emphasis on Expectation Operator E[.].
• Equation 2.44 is only one example of the use of expectation operator.
• In general, the Expectation (or Expected Value) of g(X) is given as

• The three most important expectation operators are
• Eqn 2.46 is E[X] is “mean of X” or “First moment about the origin”• Eqn 2.46 is E[X] is “mean of X” or “First moment about the origin”
• Eqn 2.47 is E[X2] is “mean of the square of X” or “Second moment
about the origin”
• Alternative notation for E[X2] is
• Eqn 2.48 is E[X-E[X]]2 is “Second moment about the mean” or
“Variance”.
• Alterative notation for Variance are

• where is called Standard Deviation

• aa

dv

Linear Algebra
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Single Random Variable:
Characteristic Functions
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Fourier Transform
Inverse Fourier Transform
Fourier Transform Pair
Inverse Fourier Transform

Linear Algebra
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Single Random variable
FUNCTIONS OF RV
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Linear Algebra
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CONDITIONED RV
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