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Random Variable

A random variable is a function defined on a sample space associated with a random experiment that assigns a real number to each element in the sample space.

 

A random variable can be discrete or continuous.

 

A random variable described on a discrete sample space such as that of tossing coins, throwing dice, deck of cards, etc. is known as a discrete random variable. Here the values are clearly defined and are finite in number.

 

A random variable described on a continuous sample space such as that of height, weight, etc. is known as a continuous random variable. Here the values are not clearly defined as the variable may take up any value in the continuous sample space.

Mathematical Expectations

Let a random variable X assume the values x1x2x3, ... xn with probabilities p1p2p3, ... pn respectively. Then the mathematical expectation E(X) will be the sum of the product of the variable values and their corresponding probabilities.

 

i.e.E(X) = p1 x1 + p2 x2 + p3 x3 + … pn xn

 

Description: 85320.png 

Mathematical expectation of a function of random variable x is given by Description: 85313.png

 

The variance of the random variable x in terms of its expectation is given by V(x) = E(x2) – [E(x)]2
 

Example
Two fair coins are tossed once. Find the mathematical expectation and the variance of the number of heads obtained.
Solution
Let X denote the number of heads obtained.
The sample space S = {TTHTTHHH}
Then, X is a random variable which takes the value 0, 1 and 2 with respective probabilities, Description: 85307.png

The mathematical expectation of the number of heads is
Description: 85275.png
Description: 85268.png 
V(x) = E(x2) – [E(x)]2
Description: 85262.png 

Important properties of expectation:

  • Expectation of a constant is a constant.
     
    i.e.E(k) = k
  • Expectation of the sum of two random variables is the sum of their expectations
     
    i.e.E(x + y) = E(x) + E(y)
  • Expectation of the product of two independent random variables is the product of their expectations
     
    i.e.E(x × y) = E(x) × E(y)
  • Expectation of the product of a constant and a random variable is the product of the constant and the expectation of the random variable
     
    i.e.E(kx) = kE(x)





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