Q 8.1. Suppose that X and Y are two random variables whose joint distribution is the standard Gaussian distribution in the plane. Let random variables R≥ 0 and = [0, 2π) satisfy X = R cos and Y = Rsin 0. (a) State the density of the joint distribution of R and O. (You do not need to derive this) (b) Express Z = in terms of and hence, making reference to part (a), calculate the density of the distribution of Z. It will help to try to picture this one.
Q 8.1. Suppose that X and Y are two random variables whose joint distribution is the standard Gaussian distribution in the plane. Let random variables R≥ 0 and = [0, 2π) satisfy X = R cos and Y = Rsin 0. (a) State the density of the joint distribution of R and O. (You do not need to derive this) (b) Express Z = in terms of and hence, making reference to part (a), calculate the density of the distribution of Z. It will help to try to picture this one.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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8.1
Transcribed Image Text:Q 8.1. Suppose that X and Y are two random variables whose joint distribution is the standard
Gaussian distribution in the plane. Let random variables R≥ 0 and = = [0, 2π) satisfy
X = R cos and Y Rsin 0.
(a) State the density of the joint distribution of R and O. (You do not need to derive this)
(b) Express Z = in terms of and hence, making reference to part (a), calculate the density
of the distribution of Z.
It will help to try to picture this one.
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for part b, could u please provide more details of the calculations I circled? thanks
Transcribed Image Text:Step3
c)
(b) Let the random variable Z =
F₂(2)=P(X ≤2)
Put
= P(Y ≤ ZX, X ≥0) + P(Y ≤ ZX, X <0)
= 2P(Y ≤ ZX, X>0)
XZ
= 25° 5x² fx(x) fy(v) dy dx
0
-∞
V
2
2
= ²5² ²2/12 - 0 + 1 = 0 ²+² oy ox
2√² √x²
e
e
dy dx
0
√√2π
√2π
∞ XZ
= 2 × ²2²1 ²²² ²²0 +² oy dx
2
2
e
e dy
2₁
0 -∞
XZ
-15 oyce why 1.00 disappears?
-22
=
e
dy dx
π
-∞
details?
= 1/1 1 0 0
e
0
2
Differentiate the function to
x²
2
(xz)²
How to diffe
√₂ (2) = = √²0 = = = (42²³²
1
2
e
e
π
0
Ell
> ×
e
Y
X
?Cound the pdf of Z,
give
и
more
- 1/²/(1+2²)
x dx
x dx
Solution
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