Starter The probability distribution of a discrete random variable X is given by: P(X = r) = 30kr for r = 3, 5, 7 P(X = r) = 0 otherwise What is the value of k? Hence or otherwise, calculate P(X ≥ 5) Calculate the area under the curve f(x) = ⅜(1 + x2) between 0 and 1.

Continuous Random Variables Learning Objectives: Understand the difference between a discrete and continuous random variable Able to determine whether a function is a probability density function Able to calculate probabilities using a p.d.f.

Discrete vs Continuous Discrete variables – can take specific values e.g. shoe size, number on a die, etc. Continuous variables – can take any value e.g. weight of a baby, height of students, time taken to run 100m, etc.

Discrete Random Variables D.R.V. – uses a probability distribution to describe the possible values

Continuous Random Variables C.R.V. – described by a probability density function (p.d.f) f(x) ≥ 0 for all x. Area under the curve must equal 1, i.e. = 1

Continuous Random Variables P(X = r) = 0 Therefore: P(a < X < b) = P(a ≤ X < b) = P(a < X ≤ b) = P(a ≤ X ≤ b)

Continuous Random Variables We can find probabilities by integrating f(x) between certain limits, i.e. P(a ≤ X ≤ b) =

Continuous Random Variables Example: A continuous random variable X has the probability density function given by f(x) = show that f(x) has the properties of a p.d.f. Find P(1.5 ≤ X ≤ 2) { ⅔x for 1 ≤ X ≤ 2 0 otherwise

f(x) ≥ 0 for all x since ⅔x > 0 for x > 0 = 1 2 1 2 1

P(1.5 ≤ X ≤ 2) = ⅔ [½x2] = ⅓ [x2] = ⅓ x (4 – 2.25) = 0.583 (3dp) 2 1.5

{ The continuous random variable X has the p.d.f. given by: f(x) = where k is a constant Find the value of k Find P(0.3 ≤ X ≤ 0.6) Find P(|X| < 0.2) { k(1 + x2) for -1 ≤ X ≤ 1 0 otherwise

{ k(1 + x2) for -1 ≤ X ≤ 1 0 otherwise f(x) = a) Find the value of k.

{ ⅜(1 + x2) for -1 ≤ X ≤ 1 0 otherwise f(x) = b) Find P(0.3 ≤ X ≤ 0.6)

{ ⅜(1 + x2) for -1 ≤ X ≤ 1 0 otherwise f(x) = c) Find P(|X| < 0.2)