Suppose that p(x)=(1)/(5), x=1,2,3,4,5, zero elsewhere, is...

Suppose that $p(x)=\frac{1}{5}, x=1,2,3,4,5$, zero elsewhere, is the pmf of the discrete-type random variable $X$. Compute $E(X)$ and $E\left(X^{2}\right)$. Use these two results to find $E\left[(X+2)^{2}\right]$ by writing $(X+2)^{2}=X^{2}+4 X+4$.

Key Concepts

Linearity of Expectation

The linearity of expectation states that the expected value of a sum of random variables (or functions thereof) is equal to the sum of their expected values, regardless of whether the variables are independent. This property is used to simplify the calculation of the expected value of a polynomial function of a random variable by allowing the separation of its terms.

Moments

Moments are expected values of powers of the random variable and provide important information about the shape of the distribution. For instance, the first moment is the mean, while the second moment is used to calculate the variance. The calculation of E(X^2) represents a second moment and is useful for further computations involving the random variable.

Probability Mass Function (PMF)

A probability mass function defines the probability that a discrete random variable is exactly equal to some value. It assigns a probability to each possible outcome in the sample space, and the sum of all probabilities must equal 1. In this context, the PMF is used to compute the expected value and other moments of the random variable.

Expected Value

The expected value (or mean) of a random variable is a measure of the central tendency of its probability distribution. It is computed as the sum of the products of each outcome and its corresponding probability. This concept is crucial for summarizing the overall behavior of a random variable.

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