Homework Answers
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8. Let f (x) e, 0 > 0; x> 0 (1 1 +e (a) Show that...
8. Let f (x) e, 0 > 0; x> 0 (1 1 +e (a) Show that f(x) is a probability density function (b) Find P(X> x) (c) Find the failure rate function of X
> + x 0. x)e-0, f(x) = fall + x)e-0x tu function. That is, shou is,...
> + x 0. x)e-0, f(x) = fall + x)e-0x tu function. That is, shou is, show that > where (a) Show that f(x) is a density fun What is f(x) > 0, and that bo f(x) dr = 1. (b) Find ELI (c) Find Var(37)
(1 point) Let f(x Scxºy? if 0 < x < 1, 0 SY51 otherwise Find the...
(1 point) Let f(x Scxºy? if 0 < x < 1, 0 SY51 otherwise Find the following: (a) c such that f(x,y) is a probability density function: c= (b) Expected values of X and Y: E(X) = E(Y) = 100 (c) Are X and Y independent? (enter YES or NO)
Let X1,...X be i.i.d with density f()(1/0)exp(-/0) for r >0 and 0> 0. a. Find the...
Let X1,...X be i.i.d with density f()(1/0)exp(-/0) for r >0 and 0> 0. a. Find the pitman estimator of 0 b. Show that the pitman estimator has smaller risk than the UMVUE of when the loss function is (t-0)2 02 Suppose C. f(x)= 0exp(-0x) and that 0 has a gamma prior with parameters a and p, find the Bayes estimator of 0 d. Find the minimum Bayes risk e. Find the minimax estimator of 0 if one exists. 1
Let...
E = "Expected Value" V = "Variance" 0 < x < 00, x < y <...
E = "Expected Value"
V = "Variance"
0 < x < 00, x < y < oo IS joint probability density function a) Compute the probability that X < 1 and Y < 2. b) Find E(X) c) Find E(Y d) Find V(X) e) Find V(Y)
1. 20 points Let X be a random variable with the following probability density function: f(x)--e+1"...
1. 20 points Let X be a random variable with the following probability density function: f(x)--e+1" with ? > 0, ? > 0, constants x > ?, (a) 5 points Find the value of constant c that makes f(x) a valid probability mass function. (b) 5 points Find the cumulative distribution function (CDF) of X.
Let f(x) = cxe-x if x 20 and f(x) = 0 if x < 0. (a)...
Let f(x) = cxe-x if x 20 and f(x) = 0 if x < 0. (a) For what value of c is fa probability density function? (b) For that value of c, find P(1<x< 4). 0.368
8. Let X = {fe (C[0, 1], || ||00): f() = 1} and Y = {fe...
8. Let X = {fe (C[0, 1], || ||00): f() = 1} and Y = {fe (C[0, 1], || |co) : 0 <f() < 1}. Show that X is complete but Y is not complete .
2x 0<x<1 Let X be a continuous random variable with probability density function f(x)= To else...
2x 0<x<1 Let X be a continuous random variable with probability density function f(x)= To else The cumulative distribution function is F(x). Find EX.
2. Let X and Y have joint density f(x.v) = \ şcy? if 0 <x< 1...
2. Let X and Y have joint density f(x.v) = \ şcy? if 0 <x< 1 and 1 <y<2, otherwise. (a) Compute the marginal probability density function of Y. If it's equal to 0 outside of some range, be sure to make this clear. (b) Set up but do not compute an integral to find P(Y < 2X).
8. Let f (x) e, 0 > 0; x> 0 (1 1 +e (a) Show that f(x) is a probability density function (b) Find P(X> x) (c) Find the failure rate function of X
> + x 0. x)e-0, f(x) = fall + x)e-0x tu function. That is, shou is, show that > where (a) Show that f(x) is a density fun What is f(x) > 0, and that bo f(x) dr = 1. (b) Find ELI (c) Find Var(37)
(1 point) Let f(x Scxºy? if 0 < x < 1, 0 SY51 otherwise Find the following: (a) c such that f(x,y) is a probability density function: c= (b) Expected values of X and Y: E(X) = E(Y) = 100 (c) Are X and Y independent? (enter YES or NO)
Let X1,...X be i.i.d with density f()(1/0)exp(-/0) for r >0 and 0> 0. a. Find the pitman estimator of 0 b. Show that the pitman estimator has smaller risk than the UMVUE of when the loss function is (t-0)2 02 Suppose C. f(x)= 0exp(-0x) and that 0 has a gamma prior with parameters a and p, find the Bayes estimator of 0 d. Find the minimum Bayes risk e. Find the minimax estimator of 0 if one exists. 1
Let...
E = "Expected Value"
V = "Variance"
0 < x < 00, x < y < oo IS joint probability density function a) Compute the probability that X < 1 and Y < 2. b) Find E(X) c) Find E(Y d) Find V(X) e) Find V(Y)
1. 20 points Let X be a random variable with the following probability density function: f(x)--e+1" with ? > 0, ? > 0, constants x > ?, (a) 5 points Find the value of constant c that makes f(x) a valid probability mass function. (b) 5 points Find the cumulative distribution function (CDF) of X.
Let f(x) = cxe-x if x 20 and f(x) = 0 if x < 0. (a) For what value of c is fa probability density function? (b) For that value of c, find P(1<x< 4). 0.368
8. Let X = {fe (C[0, 1], || ||00): f() = 1} and Y = {fe (C[0, 1], || |co) : 0 <f() < 1}. Show that X is complete but Y is not complete .
2x 0<x<1 Let X be a continuous random variable with probability density function f(x)= To else The cumulative distribution function is F(x). Find EX.
2. Let X and Y have joint density f(x.v) = \ şcy? if 0 <x< 1 and 1 <y<2, otherwise. (a) Compute the marginal probability density function of Y. If it's equal to 0 outside of some range, be sure to make this clear. (b) Set up but do not compute an integral to find P(Y < 2X).
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