Chapter 15: Random Variables

What is a random variable?

A variable that takes on numerical values determined by the outcome of a random phenomenon.

What is a discrete random variable?

A random variable with a countable number of possible values (e.g., number of heads in 10 coin flips).

What is a continuous random variable?

A random variable that can take on any value within an interval (e.g., height, weight, time).

What is a probability model for a random variable?

A table or function that shows all possible values of the variable and the probabilities that they occur.

How do you find the expected value (mean) of a random variable X?

E(X) = Σ [x × P(x)]

Multiply each value by its probability and sum the results.

How do you find the variance of a random variable X?

Var(X) = Σ [(x − μ)² × P(x)]

Where μ is the expected value (mean) of X.

How do you find the standard deviation of a random variable X?

SD(X) = √Var(X)

The square root of the variance.

If Y = a + bX, how do you find E(Y) and SD(Y)?

• E(Y) = a + bE(X)

  • SD(Y) = |b| × SD(X)

What is the rule for the mean of the sum of two random variables?

E(X + Y) = E(X) + E(Y)

Works for both independent and dependent variables.

What is the rule for the variance of the sum of two independent random variables?

Var(X + Y) = Var(X) + Var(Y)

Only applies if X and Y are independent.

Can you add standard deviations directly?

No! You must add variances first, then take the square root to get the new standard deviation.