Homework Answers
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2. Let X.... , X, N(0,%) be independent, where a > 0 is unknown. (1) Is...
Let X1,.-. , Xn ~ N(2, 1) be independent, where E R is unknown. (i) Show...
Let X1,.-. , Xn ~ N(2, 1) be independent, where E R is unknown. (i) Show that X := -1X; is a minimum sufficient statistic. (ii) Show that X is a complete statistic.
3. Let X X be a random sample from Uniform[0, where > 0 is unknown. (a)...
3. Let X X be a random sample from Uniform[0, where > 0 is unknown. (a) Show that = max{X,X is the MLE of 0. (b) Let the CDF of @ be F(-). Find F(t) for any t e R (c) Find the pdf of 0 Hint: Find the distribution function of Z maxX1,X. The first feu steps will be as follous: F2(2) P(Z) P (maxX, x) ) = P (XS2, X X,) Nert use the fact that Xis are...
Problem 1 (11 pts] The independent r.v.'s X and Y have p.d.f. f(t) = et, t>0....
Problem 1 (11 pts] The independent r.v.'s X and Y have p.d.f. f(t) = et, t>0. Compute the probability: P(X+Y > 2). Hint: Use independence of X and Y in order to find their joint p.d.f., fx,y, and then use the diagram below to compute the probability: P(X+Y < 2). y 2 r+y = 2 y . ! 2 0 2-y Note: If X and Y represent the lifetimes of 2 identical equipment of expected lifetime 1 time unit, then...
(a) Let x(t) = 1 when 0 <t<1 and 0 for all other real t. Find...
(a) Let x(t) = 1 when 0 <t<1 and 0 for all other real t. Find and graph the following: (i) r(t -3). [5] (ii) c(t/2). (5] (iii) <((t-3)/2). [5] (iv) (t/2) – 3). [5]
3. Let ,..., be independent random sample from N(), where is unknown. (i) Find a sufficient...
3. Let ,..., be
independent random sample from N(),
where is unknown.
(i) Find a sufficient statistic of .
(ii) Find the MLE of .
(iii) Find a pivotal quantity and use it to construct a
100(1–)% confidence
interval for .
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ...
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 1/μ. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt <X <Y) (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? {Z > t} = {X > t, Y > t} (e) Compute P[Z> t) wheret 0. (f) Compute the p.d.f. of Z.
(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)
7A,B,C Ar Y. Y, and : Let the joint d2 independent! Are they pairwise independent? obability...
7A,B,C
Ar Y. Y, and : Let the joint d2 independent! Are they pairwise independent? obability distribution function of X,Y, and Z be given by Flz, y, z) = (1-e-Ai*)(1-e-wa-e-A3*), x, y, z > 0, where A1, λ2, λ3 > 0. (a) Are X, Y, and Z independent? Find the joint probability density function of X, Y, and Z. (b) (c) Find P(X <Y<Z).
Let X and Y be independent and identically distributed with marginal probability density function f(a)- 0...
Let X and Y be independent and identically distributed with marginal probability density function f(a)- 0 otherwise, where 8>0 (a) [6 pts] Use the convolution formula to find the probability density function of X +Y. (b) [6 pts) Find the joint probability density function of U X+Y and V- X+Y
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).
Let X1,.-. , Xn ~ N(2, 1) be independent, where E R is unknown. (i) Show that X := -1X; is a minimum sufficient statistic. (ii) Show that X is a complete statistic.
3. Let X X be a random sample from Uniform[0, where > 0 is unknown. (a) Show that = max{X,X is the MLE of 0. (b) Let the CDF of @ be F(-). Find F(t) for any t e R (c) Find the pdf of 0 Hint: Find the distribution function of Z maxX1,X. The first feu steps will be as follous: F2(2) P(Z) P (maxX, x) ) = P (XS2, X X,) Nert use the fact that Xis are...
Problem 1 (11 pts] The independent r.v.'s X and Y have p.d.f. f(t) = et, t>0. Compute the probability: P(X+Y > 2). Hint: Use independence of X and Y in order to find their joint p.d.f., fx,y, and then use the diagram below to compute the probability: P(X+Y < 2). y 2 r+y = 2 y . ! 2 0 2-y Note: If X and Y represent the lifetimes of 2 identical equipment of expected lifetime 1 time unit, then...
(a) Let x(t) = 1 when 0 <t<1 and 0 for all other real t. Find and graph the following: (i) r(t -3). [5] (ii) c(t/2). (5] (iii) <((t-3)/2). [5] (iv) (t/2) – 3). [5]
3. Let ,..., be
independent random sample from N(),
where is unknown.
(i) Find a sufficient statistic of .
(ii) Find the MLE of .
(iii) Find a pivotal quantity and use it to construct a
100(1–)% confidence
interval for .
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 1/μ. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt <X <Y) (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? {Z > t} = {X > t, Y > t} (e) Compute P[Z> t) wheret 0. (f) Compute the p.d.f. of Z.
(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)
7A,B,C
Ar Y. Y, and : Let the joint d2 independent! Are they pairwise independent? obability distribution function of X,Y, and Z be given by Flz, y, z) = (1-e-Ai*)(1-e-wa-e-A3*), x, y, z > 0, where A1, λ2, λ3 > 0. (a) Are X, Y, and Z independent? Find the joint probability density function of X, Y, and Z. (b) (c) Find P(X <Y<Z).
Let X and Y be independent and identically distributed with marginal probability density function f(a)- 0 otherwise, where 8>0 (a) [6 pts] Use the convolution formula to find the probability density function of X +Y. (b) [6 pts) Find the joint probability density function of U X+Y and V- X+Y
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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