2. Let X · Unif(0, 1) and Y = ₁. Find the probability density function of Y. X+1' Solution. We let g: [0, 1] → [0,1/2] be given by g(x) = 41. By writing g(x) = 1+1, see that g is strictly increasing (also note that g(0) = 0 and g(1) = 1/2). The inverse function is given by | g^¹₁ (y) = 1 ² 2 y Y y = [0,1/2] 2 - (this is obtain by solving for x in y = 11). We then have fy (u) = fx (9¯¹ (u)) · |— 9¯¹ (1) = 0. (1). 7 d Y dy 1-y fx (x) = g(x) = 1 + 1 (1 — y)² ¹ y = [0,1/2]. we 1-Y Y =
2. Let X · Unif(0, 1) and Y = ₁. Find the probability density function of Y. X+1' Solution. We let g: [0, 1] → [0,1/2] be given by g(x) = 41. By writing g(x) = 1+1, see that g is strictly increasing (also note that g(0) = 0 and g(1) = 1/2). The inverse function is given by | g^¹₁ (y) = 1 ² 2 y Y y = [0,1/2] 2 - (this is obtain by solving for x in y = 11). We then have fy (u) = fx (9¯¹ (u)) · |— 9¯¹ (1) = 0. (1). 7 d Y dy 1-y fx (x) = g(x) = 1 + 1 (1 — y)² ¹ y = [0,1/2]. we 1-Y Y =
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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could u please tell me why the part I circled is 1 rather than y? thx :)
![2. Let X Unif(0, 1) and Y
=
X
X+1*
fy(y) = fx(g¯¯¹(y))
Find the probability density function of Y.
X
1 By writing g(x) =
x+1°
Solution. We let g: [0, 1] → [0, 1/2] be given by g(x) =
see that g is strictly increasing (also note that g(0)
function is given by
g¯¹ (y)
=
(this is obtain by solving for x in y
=
Y
1- y'
1
1+ X
· | d9¯¹ (0)|
dy
1
1+ 1
X
=
0 and g(1) = 1/2). The inverse
|
=(1).
Y?
y = [0, 1/2]
). We then have
fx (x) = g(x) = 1 +
1
d Y
dy 1 - y (1 — y)² ¹
- | - -
y = [0, 1/2].
we
1-Y
11
Y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb41de797-8c36-43f3-a49e-0d77bbbd163e%2F0693d2cc-8bb7-4304-bb70-cd4b897857b8%2F7ddu2v5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let X Unif(0, 1) and Y
=
X
X+1*
fy(y) = fx(g¯¯¹(y))
Find the probability density function of Y.
X
1 By writing g(x) =
x+1°
Solution. We let g: [0, 1] → [0, 1/2] be given by g(x) =
see that g is strictly increasing (also note that g(0)
function is given by
g¯¹ (y)
=
(this is obtain by solving for x in y
=
Y
1- y'
1
1+ X
· | d9¯¹ (0)|
dy
1
1+ 1
X
=
0 and g(1) = 1/2). The inverse
|
=(1).
Y?
y = [0, 1/2]
). We then have
fx (x) = g(x) = 1 +
1
d Y
dy 1 - y (1 — y)² ¹
- | - -
y = [0, 1/2].
we
1-Y
11
Y
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