(20 points) Suppose X-N(25, 81). That is, has a normal distribution with μ-25 and σ2-81. la....
(20 points) Suppose X~N(25, 81). That is, X has a normal distribution with μ-25 and σ-81 la. Find a transformation of X that will give it a mean of zero and a variance of one (ie., standardize X lb. Find the probability that 18 < χ < 26. lc. Supposing Y10 +5X, find the mean of Y ld. Find the variance ofY
1. Suppose that X ∼ Normal(μ,σ2). NOTE: The second parameter here is the variance of X, this is different to the book which uses the standard deviation as the second parameter. However, the more usual convention is to use the variance, and so that is what I will use in this class. Use the transformation technique to find and identify the distribution of (X − μ)/σ.
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and variance σ2 16. Find P (49 < Xs <51) and P (49< X <51) 2. Let Y = X1 + X2 + 15 be the sun! of a random sample of size 15 from the population whose + probability density function is given by 0 otherwise
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and...
please answer with full soultion. with explantion.
(4 points) Let Xi, , Xn denote a randon sample from a Normal N(μ, 1) distribution, with 11 as the unknown parameter. Let X denote the sample mean. (Note that the mean and the variance of a normal N(μ, σ2) distribution is μ and σ2, respectively.) Is X2 an unbiased estimator for 112? Explain your answer. (Hint: Recall the fornula E(X2) (E(X)Var(X) and apply this formula for X - be careful on the...
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...
-wa exp{-(20 )2}, where The Normal(μ,02) distribution has density f(x) -oo < μ < oo and σ > 0. Let the randon variable T be such that X-log(T) is Normal(μ, σ2). Find the density of T. This distribution is known as the log normal Do not forget to indicate where the density of T is non-zero. 10.
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t) exp ( μ1+ σ2t2 ) for -oo < t oo. Show that E X)-ψ(0)-μ and Var(X)-ψ"(0)-[ty(0)12-σ2. 5. Suppose that X1, X2, and X3 are independent random variables such that E[X]0 and ElX 1 for i-12,3. Find the value of E[LX? (2X1 X3)2] 6. Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(X, Y)- 1. Find the value of Var(3X -...
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 >...
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...
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...