Random Variables and Probability Distributions

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CHAPTER TWO
2. RANDOM VARIABLES
AND PROBABILITY
DISTRIBUTIONS

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2.1 The Concept & Definition of a Random Variable
Random Variables: - are a numerical
description of the outcome of an experiment.
The particular numerical value of the random
variable depends on the outcome of the
experiment.
 I.e. the value of the random variable is not
known until the experiment outcome is
observed.

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Cont...
Definition 2.1. If S is a sample space with a
probability measure and X is a valued function
defined over the elements of S, and then X is called
random variable.
It is a quantity resulting from an experiment that, by
chance, can assume different values.
Random variable can be classified as either
Discrete or Continuous depending on the
numerical value it assumes.

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2.2. Discrete Random Variables
Discrete Random Variables are random variables
that may assume either a finite whole number of
values or an infinite sequence of whole numbers
such as 0,1,2…is referred to as a discrete
random variable.
Although many experiments have outcomes that
are naturally described by numerical values,
others do not.

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Cont...
Example:
1. The number of students earning a grade of A in year economic
class. Here, there can be 25A s, but can not be 25.15 A grades.
‟
2. Suppose, when a coin is tossed 5 times, then the number of
heads from the given experiment will be: X = 0, 1, 2, 3, 4, 5
(i.e., if we let X be the number of heads, then the values for x
are those.)
3. For an experiment in which we roll a pair of dice say die A and
die B, observing that the random variable X takes on the value
A + B is:

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Cont...
Each of the possible out comes has the probability
1/36.
 S = {(1, 1), (1,2), (2, 1), (1, 3), (3, 1), (2, 2), (1, 4), (4,
1,), (2, 3), (3, 2), (1, 5), (5, 1), (2, 4), (4, 2), (3, 3), (1,
6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3), (2, 6), (6, 2), (3,
5), (5, 3), (4, 4), (3, 6), (6, 3), (4, 5), (5, 4), (5, 5), (4,
6), (6, 4), (5, 6), (6, 5), (6, 6)}
X 2 3 4 5 6 7 8 9 10 11 12
P(X)=X

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Cont...
Definition 2.2. If X is a discrete random variable, the
function given by f(x) = P(X=x) for each x with in the
range of X is called the probability distribution of X.
 Theorem 2.1. A function can serve as a probability
distribution of a discrete random variable, if its value
f(x), satisfy the following conditions:
 for each value within its domain.
 where the summation extends over all the values
within its domain.

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Cont...
Example 1:- Consider the experiment of rolling two
dice. Assume X is the sum of numbers shown on the
two dice. Then X will take all values as shown
below: (where X is discrete random variable).
Example 2:- For the probability of the total number
of heads obtained in 4 tosses of a coin. Describe the
probability distribution for the given experiment.
X 2 3 4 5 6 7 8 9 10 11 12
P(X)=X

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Cont...
 Solution: Let the possible out comes of heads be noted by X, the
random variable X and its corresponding probabilities by f(x), then:
 Definition 2.3. If X is a discrete random variable, the function
given for , where f(t) is the value of the probability distribution of X
at t is called the distribution function or the cumulative distribution
of X.
X 0 1 2 3 4
P(X)

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Cont...
Theorem 2.2. The values F(X) of the distribution
function of a discrete random variable X satisfy the
conditions:
1. and
2. If then for any real numbers

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Cont...
Example:- Find the distribution function of the total
number of heads obtained in five tosses of a coin.
 Solution: we get f(0) = 1/32; f(1) = 5/32; f(2) =
10/32; f(3) = 10/32; f(4) = 5/32; f(5) = 1,it follows that
by definition 2.3
 F(0) = f(0) = 1/32
 F (1) = f (0) + f(1) = 1/32 + 5/32 = 6/32 = F(0) + f(1)
=

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Cont...
 F(2) = f (0) + f(1) + f(2) = 1/32 + 5/32 + 10/32 =
16/32 = F(1) + f(2) =
 F(3) = f (0) + f(1) + f(2) + f(3) = 1/32 + 5/32 + 10/32
+ 10/32 =26/32= F(2) + f(3) =
 F(4) = f (0) + f(1) + f(2) + f(3) + f(4) = 1/32 + 5/32 +
10/32 + 10/32 + 5/32 = 31/32 + F(3) + f(4) =
 F(5) = f (0) + f(1) + f(2) + f(3) + f(4) + f(5) = 1/32 +
5/32 + 10/32 + 10/32 + 5/32+ 1/32 = 1 =F(4) + f(5) =

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Cont...
𝐹 ( 𝑥 )=
{
0 , 𝑓𝑜𝑟 𝑥 <0
1
32
, 𝑓𝑜𝑟 0 ≤ 𝑥 <1
6
32
, 𝑓𝑜𝑟 1 ≤ 𝑥 <2
16
32
, 𝑓𝑜𝑟 2 ≤ 𝑥 <3
26
32
, 𝑓𝑜𝑟 3 ≤ 𝑥 <4
31
31
, 𝑓𝑜𝑟 4 ≤ 𝑥 <5
¿ 1 , 𝑓𝑜𝑟 𝑥 ≥ 5

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Cont...
Theorem 2.3. If the range of a random variable X
consists of the values , then , and , for i = 1, 2, 3,
…,n.
 Example:- If the distribution function of X is given
by the following cumulative distribution function:
Then find,

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Cont...
𝐹 ( 𝑥 )=
{
0 , 𝑓𝑜𝑟 𝑥 <0
1
32
, 𝑓𝑜𝑟 0 ≤ 𝑥 <1
6
32
, 𝑓𝑜𝑟 1 ≤ 𝑥 <2
16
32
, 𝑓𝑜𝑟 2 ≤ 𝑥 <3
26
32
, 𝑓𝑜𝑟 3 ≤ 𝑥 <4
31
32
, 𝑓𝑜𝑟 4 ≤ 𝑥 <5
¿ 1 , 𝑓𝑜𝑟 𝑥 ≥ 5

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Cont...
Solution – Using theorem 2.3.
 f (0) = F(0) = 1/32
 f (1) = F(1) – F(0) = 6/32 – 1/32 = 5/32
 f(2) = F(2) – F(1) = 16/32 – 6/32 = 10/32
 f (3) = F(3) - F(2) = 26/32 – 16/32 = 10/32
 f (4) = F(4) – F(3) = 31/32 – 26/32 = 5/32
 f(5) = F(5) – F(4) = 32/32 – 31/32 = 1/32
 This is the total number of heads obtained in five tosses of a coin.

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2.3 Expected value and variance of discrete random variable
To summarize the behaviour of the probability
distribution, some of the most common measures include
measuring central tendency and measures of dispersion.
And these terminologies, the so called average and
dispersion are known as the expected value and the
variance of the probability distribution respectively.
2.3.1 Expected value of discrete random variable
The mean of a population is the expected value of the
population and it is denoted by E(X) or µ.

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Cont...
Therefore, the mean:
Reports the central location of the data
Is the long run average value of the random variable
in probability distribution.
Is a weighted average.
Definition 2.4. If X is discrete random variable and
f(x) is the value of its probability distribution at x, the
expected value of x is:

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Cont...
Example: A lot of 12 TV sets includes 2 with white
cords. If 3 of the 10 sets are chosen at random for
shipment to a hotel, how many sets of white cords can
the shipper expect to send to the hotel?
 Solution- X of the two sets with white cords and 3-x of
the 10 other sets can be chosen in ways, 3 of the 12
sets can be chosen in ways and, then possibilities are
equi-probable, we find that the probability distribution of
X, the number of sets with white cords shipped to the
hotel is given by:

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Cont...
, for x = 0, 1, 2.
And then,
Theorem 2.4. If X is a discrete random variable and
f(x) is the value of its probability distribution at x, the
expected value of g(x) is given by;
.
X 0 1 2
f(X)

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Cont...
Example:- If X is the number of points rolled with a
balanced die, find the expected value of
Solution: since each possible outcome has the
probability 1/6, we get:

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Cont..
 Properties of expectation
1. If a and b are constants then the expected value of
is given by:
2. if we set or , it follows from the above that;
If a is constant, then
if b is constant, then

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Cont...
3. The expectation of the sum of two functions g(x)
and h(x) is the sum of the expectations.
 I.e.,
4. If x and y are two random variables, then
5. If x and y are two independent random variables,
then:

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2.3.2 Variance of the discrete random variable
Variance is the amount of spread/variation of
distribution of a random variable.
Definition 2.5. The variance of the distribution of
X or simply the variance of X is denoted by and
is the positive square root of the variance, and is
called the standard deviation.

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 Properties of variances of a
discrete random variable
 If X is a random variable and b is a constant, then
the variance of the sum or difference of the random
variable X and the constant b is given by :
Variance is independent of change of origin.
 If a is a constant number then:
 If a and b are constant numbers then:

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Cont...
 If b is any constant number then:-
 If x and y are independent random variables, then the
variance of the sum of X and Y is given by: .
 If a and b are constants, then the variance of is given by:
In General:-
Note that if X and Y are independent.

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Cont...
Examples1: Calculate the variance of X,
representing the number of points rolled with a
balanced die.
Example 2: Calculate the variance of , representing
the number of points rolled with a balanced die.
Example 3: Calculate the variance of , for the
number of points rolled with a balanced die.

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2.4 Continuous random variables
(CRV)
Continuous random variable is a variable that can
assume one of an infinitely large number of
values. Example:-
o Height of a student
o Weight of a student
Continuous probability distribution for a
continuous variable X is called the probability
density function and is represented by a curve
bounded by two values from the abscissa and .

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Cont...
The region bounded by the two values and the
curve gives the probability that x lies between the
values a and b. I.e.
Therefore a function of the above expression can
serve as a continuous probability density function of
a continuous random variable if and only if the value
of satisfy the following conditions:

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Cont...
As in the discrete case, is called the cumulative
probability of X, then:
 ; for

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2.5 Expected value and variance of a continuous random variable
2.5.1 Expected value of a continuous random
variable
Definition 2.5. If X is a continuous random variable
and is the value of its probability density at x, the
expected value of X is:

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Cont...
Example: A certain coded measurements of an
objects diameter of threads of a fitting have the
probability density.
Then find the expected value of this random
variable.
Solution:

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Cont...
Theorem 2.5. If X is continuous random variable
and is the value of its probability density at x, the
expected value of g(x) is given by:
Example;- If X has the probability density
Find the Expected value of

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Cont...
Solution:
Properties of expectation for continuous random
variable
The properties of expectation for discrete random
variables work here too.

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Cont...
For instance: if a and b are constant and X is a
continuous random variable, then:
2.5.2 Variance of continuous random variable
Definition 2.7. The variance of the density of a
continuous random variable x is calculated by:

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Cont...
Example:- If:
Find the variance.
Solution: First find E(x). Then:

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2.6 Moment and Moment Generating Function
2.6.1 Moments: are Used to describe Characteristic
of a distribution.
Definition 2.8. The moment about the origin of a
random variable value X, denoted by is the
expected value of symbolically,
 , when x discrete and
 , when X is continuous
- is called the mean of distribution of x, or simply
the mean of x, & it is denoted by .

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Cont...
Where as the moment about the mean of a
random variable x, denoted by , is the expected
value of , symbolically,
 for r = 0, 1, 2… when x is discrete and
 when x is continuous
Note that and for any random variable for which -
exists.

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Cont...
 &
 - is called the variance of the distribution of X,
or simply the variance of x, and it is denoted by
, var(x), there fore from the above:
 for discrete random variable.
 for continuous random variable.

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2.6.2 Moment Generating Function
The moment generating function of a real valued
random variable is an alternative specification of its
probability distribution. It is a function often used to
characterize the distribution of a random variable.
The moment generating function (MGF) of a random
variable is a function of defined as:
We say that MGF of X exists, if there exists a positive
constant number , such that is finite for all .

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Cont...
Example: Let be a continuous random variable with
support and probability density function
Where is strictly positive number. The expected
value can be computed as follows;

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Cont...
 Furthermore, the above expected value exists and
is finite for any provided that . As a consequence,
passes a MGF:
 If a random variable possesses MGF , then
moment of denoted by , exists and finite for any .
That is;

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Cont...
When is the derivative of with respect to .
evaluated at the point
 Example: In the previous example we have
demonstrated that the MGF of an exponential
random variable is.

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Cont...
The expected value of can be computed by taking
the first derivative of the MGF:
And evaluated at

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Cont...
The second moment of X can be computed by
taking the second derivative of the MGF:
And evaluating it at
And so on for higher moments.

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