• Sometimes specifying the entire probability distribution is cumbersome or unnecessary.
  
• Moments of a random variable are scalar values computed from the probability distribution.
  
• They provide insight into a probability distribution’s central tendency, dispersion, skewness, kurtosis, etc.

• The expected value of a random variable is a measure of central tendency.
  
• It is the weighted average of the possible values, where the weights are the probabilities.

Discrete Random Variables

\[ \mathbb{E}(X)=\sum_{i=1}^n x_ip(x_i)=\mu_X \]

Discrete Example:

• The sample mean is an estimate of the population mean.

\[ \mathbb{E}(X)=\int_{-\infty}^{\infty} x f(x)dx=\mu_X \]

• Variance measures dispersion:

Discrete Random Variables

\[ \text{Var}(X)=\sum_{i=1}^n (x_i-\mu_X)^2 p(x_i)=\sigma_X^2 \]

• Sample statistics estimate population parameters.

• Bar graph, histogram, density plots etc are estimates of the probability distribution of a random variable.

A discrete random variable \(X\) has the following probability mass function (PMF):

Looking at the PMF plot, which of the following statements is most correct?

A. The expected value \(\mathbb{E}(X)\) is likely close to 3 because the PMF is symmetric around 3.

B. The expected value \(\mathbb{E}(X)\) is closer to 2 because lower values of \(X\) have higher probabilities than higher values.

C. The variance \(\text{Var}(X)\) is larger than 100 because there is a wide spread of values in the PMF.

D. The variance \(\text{Var}(X)\) is smaller than 100 because most of the probability mass is concentrated near the center of the distribution.