Let X be a discrete random variable, then the variance of X is :
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Step-by-Step Solution
Step 1: Define Variance
The variance of a discrete random variable X, denoted as Var(X), is defined as the expected value of the squared difference between the random variable and its mean. This measures how far the values of the random variable are spread out from the mean.
Step 2: Expand the Expectation
We can expand the squared term inside the expectation. Let E[X]=μ. Then (X−μ)2=X2−2Xμ+μ2.
Step 3: Apply Linearity of Expectation
Using the linearity property of expectation, E[aY+bZ]=aE[Y]+bE[Z], and E[c]=c for a constant c. Here, E[X] is a constant, so E[2XE[X]]=2E[X]E[X] and E[(E[X])2]=(E[X])2.
Step 4: Simplify the Expression
Combining the terms, we get the commonly used formula for the variance of a discrete random variable: the expected value of X2 minus the square of the expected value of X.