Suppose X is a random variable with possible outcomes 0 and 2, with P( X = 2 ) = 0.3. The variance of X is

The variance of a random variable is a measure of how much the values of the variable vary around the expected value. It is calculated as the sum of the squared differences between each possible outcome and the expected value, each multiplied by their respective probabilities.

Step 1: Calculate the expected value (E(X)).

The expected value is calculated as the sum of all possible outcomes each multiplied by their respective probabilities. In this case, X can take on the values 0 and 2.

Given that P(X = 2) = 0.3, this means that the probability that X = 0 must be 0.7 (because the sum of the probabilities of all possible outcomes must equal 1).

So, we can calculate E(X) as follows:

E(X) = (0 * P(X = 0)) + (2 * P(X = 2)) = (0 * 0.7) + (2 * 0.3) = 0 + 0.6 = 0.6

Step 2: Calculate the variance.

The variance is calculated as the sum of the squared differences between each possible outcome and the expected value, each multiplied by their respective probabilities.

So, the variance of X is:

Var(X) = ((0 - 0.6)^2 * P(X = 0)) + ((2 - 0.6)^2 * P(X = 2)) = ((-0.6)^2 * 0.7) + ((1.4)^2 * 0.3) = (0.36 * 0.7) + (1.96 * 0.3) = 0.252 + 0.588 = 0.84

Therefore, the variance of X is 0.84.