4.60. (a) Show that the conditional distribution function of the continuous random variable X, given a < X ≤ b, is given by F(x\a < X ≤ b) = 0 F(x) - F(a) F(b)-F(a) 1 for x≤ a for a < x≤ b for x> b (b) Differentiate the result of part (a) with respect to x to find the conditional probability density of X given a< x≤ b, and show that E[u(X)|a
4.60. (a) Show that the conditional distribution function of the continuous random variable X, given a < X ≤ b, is given by F(x\a < X ≤ b) = 0 F(x) - F(a) F(b)-F(a) 1 for x≤ a for a < x≤ b for x> b (b) Differentiate the result of part (a) with respect to x to find the conditional probability density of X given a< x≤ b, and show that E[u(X)|a
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
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Chapter1: Starting With Matlab
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4.60 Math Stats Question attached here.
4.60. (a) Show that the conditional distribution
(b) Differentiate the result of part (a) with respect to x to find the conditional probability density of X given a<X F b, and show that E[u(X)|a<X F b] = a b u(x)f(x) dx b a f(x) dx
![<
11
=
n
✪
; (x+y) for 0<x<1, 0<y<2
f(x, y) =
elsewhere
K
find the variance of W = 3X +4Y-5.
4.52. Prove Theorem 4.15.
4.53. Express var(X+Y), var(X-Y), and cov(X + Y,
X-Y) in terms of the variances and covariance of X
and Y.
4.54. If var(X₁) = 5, var(X₂) = 4, var(X3) = 7, cov(X₁,
X₂) = 3, cov(X₁, X3) = −2, and X₂ and X3 are indepen-
dent, find the covariance of Y₁ = X₁-2X₂+3X3 and
Y₂ = -2X₁ +3X2 +4X3.
4.55. With reference to Exercise 4.49, find cov(Y,Z).
4.56. With reference to Exercise 3.69 on page 100, find
the conditional mean and the conditional variance of X
given Y = -1.
4.57. With reference to Exercise 3.71 on page 100, find
the conditional expectation of the random variable U =
Z² given X = 1 and Y = 2.
4.9 The Theory in Practice
►
given A = 4.
4.59. With reference to Example 3.22 on page 94, and
part (b) of Exercise 3.78 on page 100, find the expected
value of X3X3 given X₁ = 1.
X
4.60. (a) Show that the conditional distribution function
of the continuous random variable X, given a < X ≤ b, is
given by
F(xa < X ≤ b) =
F(x) - F(a)
F(b) - F(a)
for x≤ a
for a < x≤ b
for x> b
E[u(X)|a<X ≤ b] =
Ơ
(b) Differentiate the result of part (a) with respect to
x to find the conditional probability density of X given
a< x≤ b, and show that
u(x)f(x) dx
b
[ f(x) dx
AA
140
/ 530
⠀
v
>](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F290ad8d4-4960-4767-b6d6-a911305e9e44%2F82ef543c-6569-4c16-91eb-5b819281b810%2Fck90bkid_processed.png&w=3840&q=75)
Transcribed Image Text:<
11
=
n
✪
; (x+y) for 0<x<1, 0<y<2
f(x, y) =
elsewhere
K
find the variance of W = 3X +4Y-5.
4.52. Prove Theorem 4.15.
4.53. Express var(X+Y), var(X-Y), and cov(X + Y,
X-Y) in terms of the variances and covariance of X
and Y.
4.54. If var(X₁) = 5, var(X₂) = 4, var(X3) = 7, cov(X₁,
X₂) = 3, cov(X₁, X3) = −2, and X₂ and X3 are indepen-
dent, find the covariance of Y₁ = X₁-2X₂+3X3 and
Y₂ = -2X₁ +3X2 +4X3.
4.55. With reference to Exercise 4.49, find cov(Y,Z).
4.56. With reference to Exercise 3.69 on page 100, find
the conditional mean and the conditional variance of X
given Y = -1.
4.57. With reference to Exercise 3.71 on page 100, find
the conditional expectation of the random variable U =
Z² given X = 1 and Y = 2.
4.9 The Theory in Practice
►
given A = 4.
4.59. With reference to Example 3.22 on page 94, and
part (b) of Exercise 3.78 on page 100, find the expected
value of X3X3 given X₁ = 1.
X
4.60. (a) Show that the conditional distribution function
of the continuous random variable X, given a < X ≤ b, is
given by
F(xa < X ≤ b) =
F(x) - F(a)
F(b) - F(a)
for x≤ a
for a < x≤ b
for x> b
E[u(X)|a<X ≤ b] =
Ơ
(b) Differentiate the result of part (a) with respect to
x to find the conditional probability density of X given
a< x≤ b, and show that
u(x)f(x) dx
b
[ f(x) dx
AA
140
/ 530
⠀
v
>
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