041. The Mean and the Variance of Discrete Random Variables

The Mean of probability distribution is called the Expected value of the random variable. The Expected value E(X) of a discrete random variable X is the weighted mean of all possible outcomes, in which the weights are the respective probabilities of those outcomes.

μ = E(X) = Σ[(Xi)P( xi)], (4.10)

Where Xi are the individual outcomes, P(Xi) – probability of the proper individual outcome.

Example 4.4. The expected value of the experiment of rolling a die is

μ = E(X) = [1 * 1/6] + [2 * 1/6] + [3 * 1/6] + [4 * 1/6] + [5 * 1/6] + [6 * 1/6] = 3.5

In practice, the larger number of rolls, the closer the mean is to 3.5.

The Variance of probability distribution is the mean of squared deviations from the mean. The variance σ2 may be written as

σ2 = Σ[(Xi – μ)2P(Xi)], (4.11)

Or as

σ2 = Σ[(Xi)2P(Xi)] – μ2 . (4.12)

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