(c)
So,
Y will follow normal distribution with mean 0 and variance 1, which is standard normal distribution.
** If any confusion please comment.
Assume X is a random variable following from N (μ, σ2), where σ > 0. (a)...
2. Assume X is a random variable following from N(μ, σ2), where σ > 0. (a) Write down the pdf of X (b) Compute E(X2). (b) Define YFind the distribution of Y.
Please explain very carefully! 4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 > 0 are unknown. (a) (5 marks) Let μ+σ~p denote the p-th quantile of the N(μ, σ*) distribution. What does this mean? (b) (10 marks) Determine a UMVU estimate of,1+ ơZp and justify your answer. 4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 >...
Let X be a random variable with cdf FX (x:0), expected value EIX-μ and variance VlX- σ2. Let X1,X2, , Xn be an id sample drawn according to FX(x,8) where Fx (x,8) =万 for all x E (0,0). Let max(X1, X2, , X.) be an estimator of θ, suggested from pure common sense. Remember that if Y = max(X1, X2, , Xn). Then it can be shown that the cdf Fy () of Y is given by Fr(u) (Fx()" where...
1. Suppose you are drawing a random sample of size n > 0 from N(μ, σ2) where σ > 0 is known. Decide if the following statements are true or false and explain your reasoning. Assume our 95% confidence procedure is (X- 1.96X+1.96 Vn a. If (3.2, 5.1) is a 95% CI from a particular random sample, then there is a 95% chance that μ is in this interval. b. If (3.2.5.1) is a 95% CI from a particular random...
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inean ,, and variance σ2, namely, fx (x) = exp ( 2-2 . Let X and Y be two i.i.d. random variables distributed as Gaussian with mean 0 and variance 1. Show that Z-XY is again a Gaussian random variable but with mean 0 and variance 2. Show your full proof with integrals. 2. From above, can you derive what will be the...
Suppose that X1,X2, ,Xn are iid N(μ, σ2), where both parameters are unknown. Derive the likelihood ratio test (LRT) of Ho : σ2 < σ1 versus Ho : σ2 > σ.. (a) Argue that a LRT will reject Ho when w(x)S2 2 0 is large and find the critical value to confer a size α test. (b) Derive the power function of the LRT
. Let Yi, ,Ý, be a sample from N(μ, σ2) distribution, where both μ and σ2 are un known Repeat the argument that was given in class to show that is a pivot (start by representing Yj as a linear function of a N(0, 1) random variable). Use the fact that (n-pe, of freedom") to construct the confidence interval with coverage probability 95% for σ2 (you can state the answer in terms of quantiles of X2-distribution, or find their numerical...
DISTRIBUTION OF SAMPLE VARIANCE: Xn ~ N(μ, σ2), where both μ and σ are Problem 4 (25 points). Assume that Xi unknowin 1. Using the exact distribution of the sample variance (Topic 1), find the form of a (1-0) confidence interval for σ2 in terms of quantiles of a chi-square distribution. Note that this interval should not be symmetric about a point estimate of σ2. [10 points] 2. Use the above result to derive a rejection region for a level-o...
8. Let Xi be iid N(μ, σ2) random variables. Define Y-Σ, Xi-Find the distribution of Y. a.
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...