Hadley Wickham Stat310 Continuous variables Tuesday, 3 February 2009
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Hadley Wickham
Stat310Continuous variables
Tuesday, 3 February 2009
1. Notes about the exam
2. Finish off Poisson
3. Introduction to continuous variables
4. The uniform distribution
Tuesday, 3 February 2009
• 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
Exam
Tuesday, 3 February 2009
Poisson distributionX = 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)
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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
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x
f(x)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 5 10 15 20x
f(x)
0.00
0.05
0.10
0.15
0.20
0.25
0 5 10 15 20
x
f(x)
0.00
0.05
0.10
0.15
0 5 10 15 20x
f(x)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 5 10 15 20
λ = 1 λ = 2
λ = 5 λ = 20
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What is λ?
• What is the sample space of X?
• Let’s start by looking at the mean and variance of X.
• How?
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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
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?
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Continuous random variables
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Continuous r.v.
• Sample space is the real line
• Mathematical tools: more differentiation + integration
• Same vocabulary, slightly different definitions
• New distributions
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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) ?
Intuition
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F (x) = P (X ! x) =! x
!"f(t)dt
P (X ! [a, b]) =! b
af(x)dx = F (b)" F (a)
Cumulative distribution function
P (X = a) = P (x = [a, a]) = F (a)! F (a) = 0Tuesday, 3 February 2009
f(x) For continuous x, f(x) is a probability density function.
Not a probability!
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f(x)DifferentiateIntegrate
F(x)Tuesday, 3 February 2009
f(x) ! 0 "x # R!
Rf(x) = 1
Conditions
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Questions?
Is f(x) < 1 for all x?
What do those conditions imply about F(x)?
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MX(t) =!
Retxf(x)dx
E(u(X)) =!
Ru(x)f(x)dx
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Assigns probability uniformly in an interval [a, b] of the real line
X ~ Uniform(a, b)
What are F(x) and f(x) ?
The discrete uniform
! b
af(x)dx = 1
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Intuition
X ~ Unif(1, b)
What do you expect the mean of X to be?
What about the variance?
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E(X) =a + b
2
V ar(X) =(b! a)2
12
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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
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