This document summarizes key concepts from a statistics course, including: 1. Recapping probability density functions and cumulative distribution functions. 2. Deriving the exponential distribution and its moment generating function. 3. Two techniques for transforming random variables - the distribution function technique and change of variables technique. 4. Properties of cumulative distribution functions like the memorylessness of exponential distributions.

1. Stat310 Transformations
Hadley Wickham
Monday, 16 February 2009

2. 1. Recap
2. Exponential derivation
3. Transforming random variables
1. Distribution function technique
2. Change of variables technique
4. Interesting properties of cdf
Monday, 16 February 2009

3. Recap
f (x) = cx 0 < x < 10
• What is the cdf?
• What must c be for f to be a pdf?
• What is P(2 < X < 8)?
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4. Exponential
• Derivation
• Moment generating function
Monday, 16 February 2009

5. Your turn
Let Y be the amount of time until I make a
mistake on the board. Assume Y ~
Exp(10) (i.e. I make 10 mistakes per hour).
If I go for 30 minutes without making a
mistake, what’s the probability I go for 40
minutes without making a mistake?
i.e. What is P(Y > 40 | Y > 30) ? How
does it compare to P(Y > 10)?
Monday, 16 February 2009

6. Memorylessness
• In general, if Y is exponential
• P(Y > y + a | Y > y ) = P(Y > a)
• Can you prove that?
• No memory
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7. Transformations
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8. Example
x -5 0 5 10 20
f(x) 0.2 0.1 0.3 0.1 0.3
Let X be a discrete random variable with
pmf f as defined above.
Write out the pmfs for:
A=X+2 B = 3*X C = X2
Monday, 16 February 2009

9. Continuous
Let X ~ Unif(0, 1)
What are the distributions
of the following variables?
A = 10 X
B = 5X + 3
C= X2
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10. Transformations
Distribution Change of
function variable
technique technique
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11. Distribution function
technique
X = Unif(0, 1)
Y = X2
P(Y < y) = P(X2< y) = P(X < √y)
...
Monday, 16 February 2009

12. Your turn
X ~ Exponential(θ)
Y = log(X)
Find fY(y). Does y have a named
distribution?
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13. Change of variables
If Y = u(X), and
v is the inverse of u, X = v(Y)
then
fY(y) = fX(v(y)) |v’(y)|
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14. Your turn
X ~ Exponential(θ). Y = log(X).
What is fY(y)?
X ~ Uniform(0, 10). Y = X2.
What is fY(y)?
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15. Theorem 3.5-1
IF
Y ~ Uniform(0, 1)
F a cdf
THEN
X= F -1(Y) is a rv with cdf F(x)
(Assume F strictly increasing for simplicity)
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16. Theorem 3.5-2
IF
X has cdf F
Y = F(X)
THEN
Y ~ Uniform(0, 1)
(Assume F strictly increasing for simplicity)
Monday, 16 February 2009