Download Math 426 Quiz 5 1. Consider two discrete r.v. X and Y with joint pmf

Math 426 Quiz 5
1. Consider two discrete r.v. X and Y with joint pmf given by
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f (10, 1) = f (20, 1) = f (20, 2) = 10
, f (10, 2) = f (10, 3) = 15 , f (20, 3) =
Are X and Y independent?
The joint and marginal probabilities are given in below.
3
10 .
y
1
2
3
f(x)
10
1/10
2/10
2/10
1/2
20
1/10
1/10
3/10
1/2
f(y)
1/5
3/10
1/2
x
For the pair (10, 2), we have f (10, 2) = 0.2 6= fX (10) ∗ fY (2) = 0.5 ∗ 0.3 =
0.15.
So X and Y are not independent.
2. Suppose a continue r.v. has a normal distribution with pdf fX (x) =
2
2
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√
e−(x−µ) /2σ , −∞ < x < ∞. Find the pdf of Y = eX .
2πσ
Note y = ex is an increasing function for −∞ < x < ∞.
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From y = ex , we get x = ln y, and dx
dy = y .
So the pdf of Y is given by
2
2
2
2
1
1
fY (y) = √2πσ
e−(lny−µ) /2σ y1 = √2πσy
e−(lny−µ) /2σ , y > 0.
Y is said to have a log-normal distribution because the logarithm of Y has
a normal distribution.
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3. Consider the r.v. X with pdf fX (x) = 3x2 , −1 ≤ x ≤ 1.
Find the pdf of Y = |X|. Hint: use the cdf technique.
First note y = |x| is not a monotone function of x on −1 ≤ x ≤ 1.
Ry
2
FY (y) = P (Y ≤ y) = P (|X| ≤ y) = P (−y ≤ X ≤ y) = −y 3x2 dx
=
x3 y
2 |−y
= y 3 , so fY (y) = 3y 2 , 0 ≤ y ≤ 1.
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