Chapter 16: Random Variables

Question 1

Define ‘Random variable’.

Answer 1

Assumes any of several different values as a result of some random event. Random variables are denoted by a capital letter, such as X.

Question 2

Define ‘Discrete random variable’.

Answer 2

A random variable that can take one of a finite number of distinct outcomes.

Question 3

Define ‘Continuous random variable’.

Answer 3

A random variable that can take on any of an uncountably infinite number of outcomes, typically, an interval of values on the real line.

Question 4

Define ‘Probability model or probability distribution’.

Answer 4

A function that associates a probability P with each value of a discrete random variable X, denoted P(X =x) or P(x), or with any interval of values of a continuous random variable, using a density curve.

Question 5

Define ‘Expected value’.

Answer 5

A random variable’s theoretical long-run average value, the centre of its model. Denoted μ or E(X), it is founf (if the random variable is discrete) by summing the products of variable values and their respective probabilities.
μ = E(X) = ∑xP(x)

Question 6

Define ‘Standard deviation of a random variable’.

Answer 6

Describes the spread in the model and is the square root of the variance, denoted SD(X) or σ.
σ = sqrt( Var(X) ) = sqrt( ∑ (x - μ)^2 P(x) )

Question 7

Define ‘Variance’.

Answer 7

The expected value of the squared deviations from the mean. For discrete random variables, it can be calculated as
σ^2 = Var(X) = ∑ (x - μ)^2 P(x)

Question 8

Define ‘Changing a random variable by a constant’.

Answer 8

E(X ± c) = E(X) ± c
Var(X ± c) = Var(X)
SD(X ± c) = SD(X)
E(aX) = aE(X)
Var(aX) = a^2Var(X)
SD(aX) = |a| SD(X)

Question 9

Define ‘Addition rule for expected values of random variables’.

Answer 9

E(X ± Y) = E(X) ± E(Y)

Question 10

Define ‘Addition rule for variances of random variables’. (Pythagorean Theorem of Statistics)

Answer 10

If X and Y are independent: Var(X ± Y) = Var(X) + Var(Y),
and SD(X ± Y) = sqrt( Var(X) + Var(Y) ).

Question 11

How do probability models relate probabilities to outcomes?

Answer 11

1. For discrete random variables, probability models assign a probability to each possible outcome.
2. For continuous random variables, areas under density curves assign probabilities to intervals of outcomes.