• Sometimes specififying the entire probability distribution is cumbersome or unnecessary.

• In these cases, we sometimes seek to characterize the shape of the distribution using the moments of the random variable.

• The moments of a random variable are simple scalar values that are computed from knowledge of the probability distribution.

• Moments provide insiight into a probability distribition’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 of the random variable, where the weights are the probabilities of the values.

Discrete Random Variables

\[ \begin{align} \mathbb{E}(X) &= \sum_{i=1}^n x_i p(x_i) \\ &= x_1 p(x_1) + x_2 p(x_2) + \ldots + x_n p(x_n) \\ &= \mu_X \end{align} \]

Discrete Example

• Suppose we have a random variable \(X\) with the following probability distribution:

Continuous Random Variables

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

Continuous Example

• Suppose we have a random variable \(X\) with the following probability distribution:

• The variance of a random variable is a measure of dispersion.

• It is the expected value of the squared deviation of the random variable from its mean.

Discrete Random Variables

\[ \begin{align} \text{Var}(X) &= \sum_{i=1}^n (x_i - \mu_X)^2 p(x_i) \\ &= (x_1 - \mu_X)^2 p(x_1) + (x_2 - \mu_X)^2 p(x_2) + \ldots + (x_n - \mu_X)^2 p(x_n) \\ &= \sigma_X^2 \end{align} \]

Discrete Example

• Suppose we have a random variable \(X\) with the following probability distribution:

Continuous Random Variables

\[ \begin{align} \text{Var}(X) &= \int_{-\infty}^{\infty} (x - \mu_X)^2 f(x) dx \\ &= \sigma_X^2 \end{align} \]

Continuous Example

• Suppose we have a random variable \(X\) with the following probability distribution:

• The same bar graph, histogram, or density plot are esimates of the probaility distribution of a random variable.

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

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

• The sample standard deviation is an estimate of the population standard deviation.