

P(X = x) = 1. 
all x 
Notes:
For example, if X is the rv 'the score obtained when one die is tossed', then X can takes on the values 1, 2, 3, 4, 5, 6, and therefore x = 1, 2, 3, 4, 5 or 6.
'X = 2' is read as 'the score obtained is 2', whereas 'x = 2' means 'x is 2'.
'P(X = 2)' is read as 'the probability that the score obtained is 2'.
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.
F(x) =  P(X £ 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) = 

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)] = 

Properties of E
The variance of a rv X, denoted by Var(X), is defined as
The standard deviation of X, denoted by s, is the square root of Var(X):
______  
s = Ö 

So Var(X) = s^{2}.
Computational formula for Var(X):
Properties of Var
If X and Y are any two random variables,
then for any constants a and b, If X and Y are also independent, then 
Note: Confusions often arises over the following:
2X

 twice the value of one observation of X 
X_{1} + X_{2}

 the sum of two independent obsevations of X 
E(2X) =

2E(X) 
E(X_{1} + X_{2}) =

E(X_{1}) + E(X_{2}) = 2E(X) 
Var(2X) =

2^{2}Var(X) = 4Var(X) 
Var(X_{1} + X_{2}) =

Var(X_{1}) + Var(X_{2}) = 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.
Solution:

= 1  
k(1 + 2 + ... + 6)

= 1  
k

= 1/21 
The pdf of W is P(W = w) = w/21 for w = 1, 2, ..., 6.
For w = 1, 2, ..., 6,
F(w) = 


=

(1 + ... + w)/21  
=

w(w + 1)/42 
The cdf of W is F(x) = w(w + 1)/42 for w = 1, 2, ..., 6.
E(W) = 


=

(1^{2} + 2^{2} + ... + 6^{2})/21  
=

13/3 
E(W^{2}) = 


=

(1^{3} + 2^{3} + ... + 6^{3})/21  
=

21 
Var(W) =  E(W^{2})  E^{2}(W) 
=

21  (13/3)^{2} 
=

20/9 
X = 3W  Y = W_{1} + W_{2} + W_{3} 
E(X) = 3E(W) = 13  E(Y) = E(W_{1}) + E(W_{2}) + E(W_{3}) = 13 
Var(X) = 3^{2}Var(W) = 20  Var(Y) = Var(W_{1}) + Var(W_{2}) + Var(W_{3}) = 20/3 