Post on 27-Dec-2015
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{