9 Jun 1999

andom Variables

  1. Let S be the sample space of an experiment.  A function X that assigns every s Î S a real number x is called a random variable (rv).
  2. A random variable X is a discrete random variable (drv) if the sample space S is countable, ie, S is finite or countably infinite.

P(X = x) = 1.
all x


robability Density Function

A function which is responsible for allocating probabilities of a random variable X is called the probability density function (pdf) of X.

Pdf may be given as a table or expressed as a function of x.

umulative Distribution Function

The cumulative distribution function (cdf) of a rv X is a function defined on R as follows:
F(x) =  P(X £ x)
P(X = t)
t £ x

So for a given real number x, F(x) is the sum of the probabilities up to and including the events assign to x.

Properties of F(x)


The mean value or expectation of a drv X, written as m or E(X), is given by

E(X) = 
xP(X = x).
all x

Note:  If the pdf of X is symmetrical about the central value c, then E(X) = c.

In general, if g(X) is any function of the random variable X, then

E[g(X)] = 
g(x)P(X = x).
all x

Properties of E


The variance of a rv X, denoted by Var(X), is defined as

Var(X) = E[(X - m)2].

The standard deviation of X, denoted by s, is the square root of Var(X):

s = Ö
Var(X) .

So Var(X) = s2.

Computational formula for Var(X):

Var(X) = E(X2) - [E(X)]2.

Properties of Var

um of Two Random Variables

If X and Y are any two random variables,
then for any constants a and b
E(aX ± bY) = aE(X) ± bE(Y). 

If X and Y are also independent, then 

Var(aX ± bY) = a2Var(X) + b2Var(Y).

Note:   Confusions often arises over the following:

- twice the value of one observation of X
X1 + X2
- the sum of two independent obsevations of X
E(2X) = 
E(X1 + X2) = 
E(X1) + E(X2) = 2E(X)
Var(2X) = 
22Var(X) = 4Var(X)
Var(X1 + X2) = 
Var(X1) + Var(X2) = 2Var (X). 


A die, with numbers 1 to 6, is weighted such that the probability of obtaining a score is proportional to the score.  Let W denotes the score obtained when the die is tossed once.  Find the pdf, cdf, expectation and variance of W.  If X denotes three times the score when the die is tossed once, and Y denotes the sum of the scores when the die is tossed thrice, what are the expectation and variance of X and Y.


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