This document contains notes from a statistics course covering continuous random variables. It discusses the cumulative distribution function (CDF) and probability density function (PDF) of continuous random variables. In particular, it introduces the uniform distribution and provides the formulas for its mean, variance, and how changing the support of a uniform random variable affects its distribution and variance.

1. Stat310 Continuous variables
Hadley Wickham
Tuesday, 3 February 2009

2. 1. Notes about the exam
2. Finish off Poisson
3. Introduction to continuous variables
4. The uniform distribution
Tuesday, 3 February 2009

3. Exam
• Exam structure
• Grading tomorrow
• Purpose of notes
• Question 1 - most of you managed to get
it (eventually) - at least 3 different ways
• Question 2 & 4 - did really well
• Question 3 - more of a struggle
Tuesday, 3 February 2009

4. Poisson distribution
X = Number of times some event happens
If number of events occurring in non-
overlapping times is independent, and
Probability of exactly one event occurring
in short interval of length h is ∝ λh, and
Probability of two or more events in a
sufficiently short internal is basically 0
Then X ~ Poisson(λ)
Tuesday, 3 February 2009

5. Examples
Number of calls to a switchboard
Number of eruptions of a volcano
Number of alpha particles emitted from a
radioactive source
Number of defects in a roll of paper
Tuesday, 3 February 2009

6. λ=1 λ=2
0.35 0.25
0.30
0.20
0.25
0.15
0.20
f(x)
f(x)
0.15
0.10
0.10
0.05
0.05
0.00 0.00
0 5 10 15 20 0 5 10 15 20
x x
λ=5 λ = 20
0.12
0.15
0.10
0.08
0.10
f(x)
f(x)
0.06
0.04
0.05
0.02
0.00 0.00
0 5 10 15 20 0 5 10 15 20
x x
Tuesday, 3 February 2009

7. What is λ?
• What is the sample space of X?
• Let’s start by looking at the mean and
variance of X.
• How?
Tuesday, 3 February 2009

8. What is λ?
• λ is the mean rate of events per unit
time.
• If you change the unit of time from 1 to
t, you’ll expect λt events - another
Poisson process/distribution
• ie. if X ~ Poisson(λ), and Y = tX, then Y
~ Poisson(λt)
Tuesday, 3 February 2009

9. Example
• A small amount of radioactive material
emits one alpha particle on average
every second. If we assume it is a
Poisson process, then:
• How many particles would be emitted
ever minute, on average?
• What is the probability that no particles
are emitted in 10 seconds?
Tuesday, 3 February 2009

10. Continuous random
variables
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11. Continuous r.v.
• Sample space is the real line
• Mathematical tools: more differentiation
+ integration
• Same vocabulary, slightly different
definitions
• New distributions
Tuesday, 3 February 2009

12. Intuition
Imagine you have a spinner which is
equally likely to point in any direction. Let
X be the angle the spinner points.
What is P(X ∈ [0, 90]) ? What is P(X ∈
[270, 90]) ? What is P(X ∈ [70, 98]) ?
What is the general formula?
What is P(X = 90) ?
Tuesday, 3 February 2009

13. Cumulative distribution function
x
F (x) = P (X ≤ x) = f (t)dt
−∞
b
P (X ∈ [a, b]) = f (x)dx = F (b) − F (a)
a
P (X = a) = P (x = [a, a]) = F (a) − F (a) = 0
Tuesday, 3 February 2009

14. f (x) For continuous x,
f(x) is a probability
density function.
Not a probability!
Tuesday, 3 February 2009

15. f (x)
Integrate Differentiate
F(x)
Tuesday, 3 February 2009

16. Conditions
f (x) ≥ 0 ∀x ∈ R
f (x) = 1
R
Tuesday, 3 February 2009

17. Questions?
Is f(x) < 1 for all x?
What do those conditions imply about
F(x)?
Tuesday, 3 February 2009

18. E(u(X)) = u(x)f (x)dx
R
MX (t) = e f (x)dx
tx
R
Tuesday, 3 February 2009

19. The discrete uniform
Assigns probability uniformly in an interval
[a, b] of the real line
X ~ Uniform(a, b)
What are F(x) and f(x) ?
b
f (x)dx = 1
a
Tuesday, 3 February 2009

20. Intuition
X ~ Unif(1, b)
What do you expect the mean of X to be?
What about the variance?
Tuesday, 3 February 2009

21. a+b
E(X) =
2
(b − a)
2
V ar(X) =
12
Tuesday, 3 February 2009

22. Question
X ~ Unif(0, 1)
Y = 10 X
What is the distribution of Y?
How does the variance of Y compare to
the variance of X?
Tuesday, 3 February 2009