Discrete Random Variables
A random variable which takes finite or countably finite values is called a Discrete random variable.
Ex. Two fair dice are thrown simultaneously. Let X= Sum of two dice.
Then X is a discrete random variable which takes the values 2, 3, 4…….12.
 Probability Distributions and Probability Mass Functions
Probability Distributions and Probability Mass Functions


Figure: Probability distribution for bits in error.
Probability Distributions and Probability Mass Functions


Figure: Loadings at discrete points on a long, thin beam.
 Cumulative Distribution Functions
Cumulative Distribution Functions
F(x) is a non decreasing function
F(x) is a step function 
Example: Let X is a discrete random variable with the following probability mass function. Find the cumulative distribution function and plot on the graph.
x
0
1
2
P(x)
0.886
0.111
0.003
Solution
Figure  Cumulative distribution function  
X
1
2
3
P(x)
0.886
0.111
0.003
F(x)
0.886
0.997
1.000
Examples
Find the cumulative distribution function of the random variable representing single dice roll.
Solution:  
X
1
2
3
4
5
6
P(x)
1/6
1/6
1/6
1/6
1/6
1/6
F(x)
1/6
2/6
3/6
4/6
5/6
6/6
F(x) = 0 if x < 1
        = 1/6 if 1 ≤  x < 2
         = 2/6 if 2 ≤ x <3
         = 3/6 if 3 ≤ x < 4
          = 4/6 if 4 ≤ x < 5
          = 5/6 if 5 ≤ x < 6
          = 6/6 = 1 if x ≥ 6
               
Examples
From the following probability mass function find c.d.f. Also plot in the graph.


Also find P(-1< X ≤ 2), P(1 ≤ X ≤ 5), P(0<X<2), P(X = even)

A box contains 4 blue and 6 red balls. Four balls are drawn at random. Obtain probability mass function and c.d.f.
X
-3
-1
0
1
2
3
5
8
P(x)
0.1
0.2
0.15
0.2
0.1
0.15
0.05
0.05
Mean of a Discrete Random Variable
Variance of a Discrete Random Variable
Mean and Variance of a Discrete Random Variable

A probability distribution can be viewed as a loading with the mean equal to the balance point.  Parts (a) and (b) illustrate equal means, but Part (a) illustrates a larger variance.
 Mean and Variance of a Discrete Random Variable

 The probability distribution illustrated in Parts (a) and (b) differ even though they have equal means and equal variances.

Example:1
Let X be the discrete random variable that takes the values 1, 2, 4, 8, and 16, each with probability 1/5. Compute the expectation and Variance of X.
Find Expectation and Variance of the number on a dice when thrown.
Find Expectation of the sum of numbers when two unbiased dice are thrown.



Examples: 
Properties of Expectation
E((f(X) + a) = E(f(X)) + a

E(aX + b) = aE(X) + b

If X>0, then E(X)>0

If X and Y are two random variables such that X<Y, then E(X)<E(Y)
Properties of Expectation
Properties of Variance
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