Post on 25-May-2015
Hadley Wickham
Stat310Continuous random variables
Thursday, 4 February 2010
1. Finish off Poisson
2. New conditions & definitions
3. Uniform distribution
4. Exponential and gamma
Thursday, 4 February 2010
Challenge
If X ~ Poisson(109.6), what is P(X > 100) ?
What’s the probability they score over 100 points? (How could you work this out?)
Thursday, 4 February 2010
grid
cumulative
0.0
0.2
0.4
0.6
0.8
1.0
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80 100 120 140
Thursday, 4 February 2010
Multiplication
X ~ Poisson(λ)
Y = tX
Then Y ~ Poisson(λt)
Thursday, 4 February 2010
Continuous random variables
Recall: have uncountably many outcomes
Thursday, 4 February 2010
f(x) For continuous x, f(x) is a probability density function.
Not a probability!(may be greater than one)
Thursday, 4 February 2010
P (a < X < b) =� b
af(x)dx
What does P(X = a) = ?
P (a ≤ X ≤ b) =� b
af(x)dx
Thursday, 4 February 2010
f(x) ≥ 0 ∀x ∈ R�
Rf(x) = 1
Conditions
Thursday, 4 February 2010
MX(t) =�
Retxf(x)dx
E(u(X)) =�
Ru(x)f(x)dx
Thursday, 4 February 2010
AsideWhy do we need different methods for discrete and continuous variables?
Our definition of integration (Riemanian) isn’t good enough, because it can’t deal with discrete pdfs.
Can generalise to get Lebesgue integration. More general approach called measure theory.
Thursday, 4 February 2010
F (x) = P (X ≤ x) =� x
−∞f(t)dt
P (X ∈ [a, b]) =� b
af(x)dx = F (b)− F (a)
Cumulative distribution function
Thursday, 4 February 2010
f(x)DifferentiateIntegrate
F(x)Thursday, 4 February 2010
Uniform distribution
Thursday, 4 February 2010
Assigns probability uniformly in an interval [a, b] of the real line
X ~ Uniform(a, b)
What are F(x) and f(x) ?
The uniform
� b
af(x)dx = 1
Thursday, 4 February 2010
Intuition
X ~ Unif(a, b)
How do you expect the mean to be related to a and b?
How do you expect the variance to be related to a and b?
Thursday, 4 February 2010
f(x) =1
b− ax ∈ [a, b]
F (x) = 0 x < a
F (x) = x−ab−a x ∈ [a, b]
F (x) = 1 x > b
Thursday, 4 February 2010
integrate e^(x * t) 1/(b-a) dx from a to b
MX(t) = E[eXt] =� b
aext 1
b− adx
Thursday, 4 February 2010
Mean & varianceFind mgf: integrate e^(x * t) 1/(b-a) dx from a to b
Find 1st moment: d/dt (E^(a_1 t) - E^(b_1 t))/(a_1 t - b_1 t) where t = 0
Find 2nd moment d^2/dt^2 (E^(a_1 t) - E^(b_1 t))/(a_1 t - b_1 t) where t = 0
Find var: simplify 1/3 (a_1 b_1+a_1^2+b_1^2) - ((a_1 + b_1)/2) ^ 2
Thursday, 4 February 2010
E(X) =a + b
2
V ar(X) =(b− a)2
12
Thursday, 4 February 2010
Write out in a similar way when using wolfram
alpha in homework
Thursday, 4 February 2010
Exponential and gamma distributions
Thursday, 4 February 2010
Conditions
Let X ~ Poisson(λ)
Let Y be the time you wait for the next event to occur. Then Y ~ Exponential(β = 1 / λ)
Let Z be the time you wait for α events to occur. Then Z ~ Gamma(α, β = 1 / λ)
Thursday, 4 February 2010
LA Lakers
Last time we said the distribution of scores looked pretty close to Poisson. Does the distribution of times between scores look exponential?
(Downloaded play-by-play data for all NBA games in 08/09 season, extracted all point scoring plays by Lakers, computed times between them - 4660 in total)
Thursday, 4 February 2010
value
Proportion
0.0000.0050.0100.0150.0200.0250.030
0.0000.0050.0100.0150.0200.0250.030
0.0000.0050.0100.0150.0200.0250.030
1
4
7
0 50 100 150 200 250 300
2
5
8
0 50 100 150 200 250 300
3
6
9
0 50 100 150 200 250 300
8 are exponential. 1 is real.
Thursday, 4 February 2010
dur
Proportion
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 50 100 150 200 250 300
What causes this spike?
Thursday, 4 February 2010
dur
Proportion
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 50 100 150 200 250 300
β = 44.9 s
Thursday, 4 February 2010
Why?
Why is it easy to see that the waiting times are not exponential, when we couldn’t see that the scores were not Poisson?
Thursday, 4 February 2010
Next
Remove free throws.
Try using a gamma instead (it’s a bit more flexible because it has two parameters, but it maintains the basic idea)
Thursday, 4 February 2010
value
Proportion
0.000
0.005
0.010
0.015
0.020
0.000
0.005
0.010
0.015
0.020
0.000
0.005
0.010
0.015
0.020
1
4
7
0 100 200 300 400
2
5
8
0 100 200 300 400
3
6
9
0 100 200 300 400
8 are gamma. 1 is real.
Thursday, 4 February 2010
dur
Proportion
0.000
0.005
0.010
0.015
0.020
0 100 200 300 400
Thursday, 4 February 2010
dur
Proportion
0.000
0.005
0.010
0.015
0.020
0 100 200 300 400
β = 28.3α = 2.28
Out of 6611 shots, 3140 made it:
1 went in for every 2.1 attempts
Thursday, 4 February 2010
f(x) =1θe−x/θ
M(t) =1
1− θt
Exponential Gamma
mgf
Mean
Variance
θ
θ2
f(x) =1
Γ(α)θαxα−1e−x/θ
M(t) =1
(1− θt)α
αθ
αθ2
Thursday, 4 February 2010
Another derivation
Find mgf: integrate 1 / (gamma(alpha) theta^alpha) x^(alpha - 1) e^(-x / theta) e^(x*t) from x = 0 to infinity
Find mean: d/dt theta^(-alpha) (1/theta - t)^(-alpha) where t = 0
Rewrite: d/dt x_2^(-x_1) (1/x_2 - t)^(-x_1) where t = 0
Thursday, 4 February 2010
Your turn
Assume the distribution of offensive points by the LA lakers is Poisson(109.6). A game is 48 minutes long.
Let W be the time you wait between the Lakers scoring a point. What is the distribution of W?
How long do you expect to wait?
Thursday, 4 February 2010
Next week
Normal distribution: Read 3.2.4
Calculating probabilities
Transformations: Read 3.4 up to 3.4.3
Thursday, 4 February 2010