![We are given X ~ Poisson(X) and Y ~ Uniform(a,b) sampled IID from known distributions. We define: (a) What is E[A] and E[B]? (b) What is var(A) and var(B)? Helpful identity. EX2-λ2 + λ (c) What is covariance and what is correlation? Explain in your own words. (d) What is the covariance of A and B? (e) Define independence of two random variables. Are A and B independent? Explain.](http://img.homeworklib.com/questions/c4a13110-64b5-11eb-94eb-2b6526ca293f.png?x-oss-process=image/resize,w_560)
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We are given X ~ Poisson(X) and Y ~ Uniform(a,b) sampled IID from known distributions. We...
Only ques 4 (b) Define R = X(n)-X(1) as the sample range. Find the pdf of...
Only ques 4
(b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , Xn ~ (iid) Uniform(0,0), E(R)-θ . What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively. 4. Let X. Xn be a random sample from a population with pdf xotherwise Let Xa)<..< X(n) be the order statistics. Show that Xa)/X() and X(n) are independent random variables 5....
3. Again, let XXn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf...
3. Again, let XXn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xo) and X(a) (b) Define R-X(n) - Xu) as the sample range. Find the pdf of R (c) It turns out, if Xi, X, n(iid) Uniform(0,e), E(R)- What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively.
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability...
Recall that a discrete random variable X has Poisson
distribution with parameter λ if the probability mass function of
X
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf...
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xi) and X(n)- (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , xn (iid) Uniform(0,0), E(R)-θ . What happens to E® as n increases? Briefly explain in words why this makes sense intuitively.
b) Show that x, -x)-o a) Suppose Y =-X . Show in a diagram this function,...
b) Show that x, -x)-o a) Suppose Y =-X . Show in a diagram this function, what will be the correlation coefficient between X and Y? 4 the correlation coefficient must b) i) If the covariance between two variables is be positive. True or False? suggest? i) If the covariance between two variables is zero, what does it 5 a) Define mutually exclusive events and independent events bi) For two events A and B (which are not mutually exclusive) complete...
you have two random variables, X and Y with joint distribution given by the following table:...
you have two random variables, X and Y with joint distribution given by the following table: Y=0 | .4 .2 4+.26. So, for example, the probability that Y 0, X - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),f(r). (b) Find the conditional distribution (pmf) of Y give X, denoted f(Y|X). (c) Find the expected values of X and Y, E(X), E(Y). (d) Find the variances of X...
1. Suppose you have two random variables, X and Y with joint distribution given by the...
1. Suppose you have two random variables, X and Y with joint distribution given by the following tables So, for example, the probability that Y o,x - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),J(Y). (b) Find the conditional distribution (pmf) of Y give X, denoted f(YX). (c) Find the expected values of X and Y, EX), E(Y). (d) Find the variances of X and Y, Var(X),Var(Y). (e)...
Suppose that X, Y and Z are all independent of each other, with the following distributions:...
Suppose that X, Y and Z are all independent of each other, with the following distributions: X Poisson(1) Y ~ Gamma(a,b) ZN(0,1) Define A as the sum: A = X+Y+Z a What is E[A]? b What is the MGF of A? (you don't need to re-derive the individual mgfs) c Use mA(t) to find E[A] (should match part a)
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y...
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y ~ N(5,9) (we use the notation N (?, ?. ) ). Let W 3X-2Y + 1. (a) Compute E(W) and Var(W) (b) It is known that the sum of independent normal distributions is n Compute P(W 6)
(20 points) Consider the following joint distribution of X and Y ㄨㄧㄚ 0 0.1 0.2 1 0.3 0.4 (a) Find the marginal distributions of X and Y. (i.e., Px(x) and Py()) (b) Find the conditional distributio...
(20 points) Consider the following joint distribution of X and Y ㄨㄧㄚ 0 0.1 0.2 1 0.3 0.4 (a) Find the marginal distributions of X and Y. (i.e., Px(x) and Py()) (b) Find the conditional distribution of X given Y-0. (i.e., Pxjy (xY-0)) (c) Compute EXIY-01 and Var(X)Y = 0). (d) Find the covariance between X and Y. (i.e., Cov(X, Y)) (e) Are X and Y independent? Justify your answer.
(20 points) Consider the following joint distribution of X and...
Only ques 4
(b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , Xn ~ (iid) Uniform(0,0), E(R)-θ . What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively. 4. Let X. Xn be a random sample from a population with pdf xotherwise Let Xa)<..< X(n) be the order statistics. Show that Xa)/X() and X(n) are independent random variables 5....
3. Again, let XXn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xo) and X(a) (b) Define R-X(n) - Xu) as the sample range. Find the pdf of R (c) It turns out, if Xi, X, n(iid) Uniform(0,e), E(R)- What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively.
Recall that a discrete random variable X has Poisson
distribution with parameter λ if the probability mass function of
X
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xi) and X(n)- (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , xn (iid) Uniform(0,0), E(R)-θ . What happens to E® as n increases? Briefly explain in words why this makes sense intuitively.
b) Show that x, -x)-o a) Suppose Y =-X . Show in a diagram this function, what will be the correlation coefficient between X and Y? 4 the correlation coefficient must b) i) If the covariance between two variables is be positive. True or False? suggest? i) If the covariance between two variables is zero, what does it 5 a) Define mutually exclusive events and independent events bi) For two events A and B (which are not mutually exclusive) complete...
you have two random variables, X and Y with joint distribution given by the following table: Y=0 | .4 .2 4+.26. So, for example, the probability that Y 0, X - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),f(r). (b) Find the conditional distribution (pmf) of Y give X, denoted f(Y|X). (c) Find the expected values of X and Y, E(X), E(Y). (d) Find the variances of X...
1. Suppose you have two random variables, X and Y with joint distribution given by the following tables So, for example, the probability that Y o,x - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),J(Y). (b) Find the conditional distribution (pmf) of Y give X, denoted f(YX). (c) Find the expected values of X and Y, EX), E(Y). (d) Find the variances of X and Y, Var(X),Var(Y). (e)...
Suppose that X, Y and Z are all independent of each other, with the following distributions: X Poisson(1) Y ~ Gamma(a,b) ZN(0,1) Define A as the sum: A = X+Y+Z a What is E[A]? b What is the MGF of A? (you don't need to re-derive the individual mgfs) c Use mA(t) to find E[A] (should match part a)
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y ~ N(5,9) (we use the notation N (?, ?. ) ). Let W 3X-2Y + 1. (a) Compute E(W) and Var(W) (b) It is known that the sum of independent normal distributions is n Compute P(W 6)
(20 points) Consider the following joint distribution of X and Y ㄨㄧㄚ 0 0.1 0.2 1 0.3 0.4 (a) Find the marginal distributions of X and Y. (i.e., Px(x) and Py()) (b) Find the conditional distribution of X given Y-0. (i.e., Pxjy (xY-0)) (c) Compute EXIY-01 and Var(X)Y = 0). (d) Find the covariance between X and Y. (i.e., Cov(X, Y)) (e) Are X and Y independent? Justify your answer.
(20 points) Consider the following joint distribution of X and...
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