MEAN AND VARIANCE OF A RANDOM VARIABLE - Quantitative Techniques for management

The mean and variance of a random variable can be computed in a manner similar to the computation of mean and variance of the variable of a frequency distribution.

Mean: If X is a discrete random variable which can take values X1, X2, ..... Xn, with respective probabilities as p(X1), p(X2), ...... p(Xn), then its mean, also known as the Mathematical Expectation or Expected Value of X, is given by :

Mathematical Expectation or Expected Value of X

The mean of a random variable or its probability distribution is often denoted by m , i.e., E(X) = m .

Remarks: The mean of a frequency distribution can be written as

mean of a frequency distribution

which is identical to the expression for expected value.

Variance: The concept of variance of a random variable or its probability distribution is also similar to the concept of the variance of a frequency distribution. The variance of a frequency distribution is given by

expression for expected value

The expression for variance of a probability distribution with mean m can be written in a similar way, as given below :

expression for expected value

where X is a discrete random variable.

Remarks: If X is a continuous random variable with probability density function p(X),

Then

continuous random variable with probability density function

Moments

The rth moment of a discrete random variable about its mean is defined as:

moment of a discrete random variable

Similarly, the rth moment about any arbitrary value A, can be written as

rth moment about any arbitrary value A

The expressions for the central and the raw moments, when X is a continuous random variable, can be written as

continuous random variable


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