Random Variable - Mean and Variance

Random Variable: We all know about a random experiment while
studying the theory of probability. To every outcome of random variable
a real number can be associated. Random variable denotes nothing but
this correspondence between the elements of sample space associated
with the random experiement with the set of real numbers. A particular
outcome of random variable is called random variate.

Depending upon the type of value assumed by it the random variable is divided into two types:

1.Discrete random variable: When the random variable assumes values which are countable, the random variable is called discrete.

2.Continuous
random variable: When the random variable assumes values which are
continuous, the random variable is called continuous.


Probability Distribution: Now, suppose a random variables assumes values x₁, x₂, x₃ .... x_n with probability of each value being p₁, p₂, p₃ ....... p_n respectively. Then we can define the probability distribution as:

X:              x₁    x₂    x₃    ................    x_n

P(X):          p₁    p₂    p₃    ...............    p_n

It is nothing but a table giving the probability of occurence of each value of the random variable.

The


Remember: The probability of each value of random variable must satisfy: Σp_i = 1

Let us take an example.


Example: Three cards are drawn from a pack of 52 cards.Find the probability distribution of the number of the aces. [CBSE 2001]

Solution: P (X = 0) Probability of getting no ace = ⁴⁸C₃/⁵²C₃  =    4324/5525
               P (X = 1) Probability of getting one ace = ⁴⁸C₂ X ⁴C₁/⁵²C₃ = 1128/5525
               P (X = 2) Probability of getting two aces = ⁴⁸C₁ X ⁴C₂/⁵²C₃ = 72/5525
               P (X = 3) Probability of getting three aces = ⁴C₃/⁵²C₃  = 1/5525


Mean of a discrete random variable:
Mean of a random variable {also called expected value or mathematical
expectation E(X)} is the mean of its probability distribution. Suppose
a discrete random variable assumes values x₁, x₂, x₃ .... x_n and their respective probabilities are p₁,p₂,p₃ ..... p_n, then the mean of this random variable is given by:



Note: In case of frequency distribution, the proabability of each value can be calculated from its frequency.
i.e, p_i = f_i / (f₁ + f₂ + f₃ .............. + f_n) = f_i / N .

Example:
Suppose a insurance agent sells life insurance to people and the
probability of selling the various number of insuance in a day as seen
from his performance sheet is given by:

Numbers sold in a day:   0        1        2        3        4        5
Probability:                   0.1     0.2      0.25    0.2    0.15     0.1

Solution:   


Variance of a discrete random variable: If X is a discrete random variable which takes values x₁, x₂, x₃ .... x_n, with p₁, p₂, p₃ ....... p_n respectively being their probabilities then the variane X is defined as: