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bd40bc7c7a We see that 2(1-x) = 2 - 2x ≥ 0 precisely when x ≤ 1; thus f(x) is everywhere nonnegative. So f(x) is indeed a valid PDF. Part 1. Using this value, we compute the variance of X as follows . There is an alternative formula for the variance of a random variable that is less tedious than the above definition. To verify that f(x) is a valid PDF, we must check that it is everywhere nonnegative and that it integrates to 1. .. Alternate Formula for the Variance of a Continuous Random Variable The variance of a continuous random variable X with PDF f(x) is the number given by . Calculating the variance of X requires its expected value: .
An Alternative Formula for Variance. We should note that a completely analogous formula holds for the variance of a discrete random variable, with the integral signs replaced by sums. To check that f(x) has unit area under its graph, we calculate . Check that this is a valid PDF and calculate the standard deviation of X. Solution. Therefore, the standard deviation of X is . The derivation of this formula is a simple exercise and has been relegated to the exercises. The random variable X is given by the following PDF. Simple Example. Part 2.