IFT Notes for Level I CFA^® Program

R09 Common Probability Distributions

Part 2

3. Continuous Random Variables

3.1. Continuous Uniform Distribution

The continuous uniform distribution is defined over a range from a lower limit ‘a’ to an upper limit ‘b’. These limits serve as the parameters of the distribution.

The probability that the random variable will take a value between x₁ and x₂, where x₁ and x₂ both lie within the range is given by:

P(x₁ ≤ X ≤ x₂) = $\frac{x_2\ - \ x_1}{b \ - \ a}$

Example

X is a uniformly distributed continuous random variable between 10 and 20. Calculate the probability that X will fall between 12 and 18.

Solution:

P(12 ≤ X ≤ 18) = $\frac{ 18\ - \ 12}{ 20 \ - \ 10} = 0.6$

The cumulative distribution function for a continuous random variable is shown below:

Example

A commodity analyst predicts that the price per ounce of gold three years from now will be between $1,500 and $1,700. Assume gold prices follow a continuous uniform distribution. What is the probability that the price will be less than $1,600 three years from now?

Solution:

F(1,600) = $\frac{1600 - 1500}{1700 - 1500} = 50\%$ . The probability that gold price will be less than $1,600 per ounce three years from now is 50%.