Discrete Random Variables: Expected Value and Moments

Discrete Random Variables

Expected Value and Higher Moments

The expected value (mean) of a discrete random variable provides a measure of the central tendency or average outcome. Moments are statistical measures that describe the shape and spread of the distribution.

• Expected Value (Mean): For a discrete random variable X with possible values xi and probabilities P(X = xi), the expected value is given by: $\mu = E[X] = \sum_i x_i P(X = x_i)$

• kth Raw Moment: The kth raw moment is defined as: $E[X^k] = \sum_i (x_i)^k P(X = x_i)$

• kth Central Moment: The kth central moment measures the spread around the mean: $E[(X - \mu)^k] = \sum_i (x_i - \mu)^k P(X = x_i)$

Example: Calculating Expected Value and Moments

Suppose X is a discrete random variable with the following probability mass function:

x	P(X = x)
1	0.4
2	0.2
3	0.3
4	0.1

• Expected Value: $E[X] = 1 \times 0.4 + 2 \times 0.2 + 3 \times 0.3 + 4 \times 0.1 = 2.1$

• Second Moment: $E[X^2] = 1^2 \times 0.4 + 2^2 \times 0.2 + 3^2 \times 0.3 + 4^2 \times 0.1 = 5.1$

• First Central Moment: $E[X - \mu] = 0$ (always zero for any random variable)

• Second Central Moment (Variance): $E[(X - \mu)^2] = \sum_i (x_i - 2.1)^2 P(X = x_i)$

Expected Value of Transformed Random Variables

When a random variable X is transformed by a function g(X), the expected value of the new variable Y = g(X) is calculated as:

• $\mu_Y = E[Y] = E[g(X)] = \sum_i g(x_i) P(X = x_i)$

Linear Property of Expected Values

The expected value operator is linear, meaning it distributes over addition and scalar multiplication:

• If X and Y are random variables and a, b are constants: $E[aX + bY] = aE[X] + bE[Y]$

Example: Linear Transformation

If $E(X) = 5$ and $Y = 3X + 2$, then:

• $E(Y) = E[3X + 2] = 3E[X] + 2 = 3 \times 5 + 2 = 17$

Example: Expected Value of a Nonlinear Transformation

Let X have the following probability distribution:

x	f(x)
-3	1/12
6	1/2
9	5/12

Find $\mu_{g(X)}$ where $g(X) = (3X + 3)^2$:

• Calculate $g(x)$ for each x, multiply by f(x), and sum.

Example: Variance Calculation

The variance of a random variable X measures the spread of its values:

• Given the probability distribution:

x	f(x)
2	0.04
3	0.01
4	0.40
5	0.25
6	0.30

• Variance: $Var(X) = E[(X - \mu)^2]$

Example: Calculating E(X), E(X2), and E[(3X+1)2]

Let X have the following probability distribution:

x	f(x)
-2	1/3
4	1/12
6	7/12

• Calculate $E(X)$ and $E(X^2)$ using the formulas above.

• To find $E[(3X + 1)^2]$, expand the expression and use linearity:

• $E[(3X + 1)^2] = E[9X^2 + 6X + 1] = 9E[X^2] + 6E[X] + 1$

Summary Table: Key Formulas for Discrete Random Variables

Concept	Formula
Expected Value	$E[X] = \sum_i x_i P(X = x_i)$
kth Raw Moment	$E[X^k] = \sum_i (x_i)^k P(X = x_i)$
kth Central Moment	$E[(X - \mu)^k] = \sum_i (x_i - \mu)^k P(X = x_i)$
Variance	$Var(X) = E[(X - \mu)^2]$
Expected Value of Transformation	$E[g(X)] = \sum_i g(x_i) P(X = x_i)$
Linearity	$E[aX + bY] = aE[X] + bE[Y]$

Additional info: The notes cover the core concepts of discrete random variables, expected value, moments, variance, and transformations, which are directly relevant to college-level statistics courses.