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 variableAble to determine whether a function is a probability density functionAble 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) =

(a) show that f(x) has the properties of a p.d.f.(b) Find P(1.5 ≤ X ≤ 2)

⅔x for 1 ≤ X ≤ 20 otherwise{

• f(x) ≥ 0 for all x since ⅔x > 0 for x > 0

= ⅔ [½x2]

= ⅓ [x2]

= ⅓ x (4 – 1)

= 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

2

1.5

• The continuous random variable X has the p.d.f. given by:

f(x) =

where k is a constant

a) Find the value of kb) Find P(0.3 ≤ X ≤ 0.6)c) Find P(|X| < 0.2)

k(1 + x2) for -1 ≤ X ≤ 10 otherwise{

• f(x) =

a) Find the value of k.

k(1 + x2) for -1 ≤ X ≤ 10 otherwise{

• f(x) =

b) Find P(0.3 ≤ X ≤ 0.6)

⅜(1 + x2) for -1 ≤ X ≤ 10 otherwise{

• f(x) =

• c) Find P(|X| < 0.2)

⅜(1 + x2) for -1 ≤ X ≤ 10 otherwise{