value at risk at level p, p is
between 0 to 1
Homework Answers
value at risk at level p, p is between 0 and 1 Q2. Denote by Ф...
value at risk at level p, p is between 0 and 1
Q2. Denote by Ф the standard normal distribution (that is, with mean zero and variance 1), and by ф_l the corresponding quantile function. Let X be a normal random variable with mean μ and variance σ2. Evaluate VaRx (p) in terms of ф-1, , 1 and σ. Hint: X = ơZ + μ, where Z is standard normal
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inea...
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...
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 > 0 are unk...
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 >...
(4 points) Let Xi, , Xn denote a randon sample from a Normal N(μ, 1) distribution, with 11 as the...
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...
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and...
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and variance σ , with known μ and unknown σ' . You are given that Σ(X-μ)2 is sufficient for σ a) Find El Σ(X-μ). |. Show all steps. Use the fact that: Var(Y)-E(P)-(BY)' i-1 b) Find the MVUE of σ.
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased...
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased estimator for the common standard deviation of two independent sam ples Let Yi, Yi2, ..., Yini denote the random sample of size n from the first population with population mean μ| and population variance σ, and let Y21, Y22, ..., Y2na denote an independent random sample of size n2 from the second population with population mean μ2 and population mean ơ3. Sup- pose that...
QUESTION： Yi, Y2, Y, denote a random sample from the normal distribution with known mean μ...
QUESTION：
Yi, Y2, Y, denote a random sample from the normal distribution with known mean μ 0 and unknown variance σ 2, find t 1 he method-of-moments estimator of σ 2 C2. Continue with Exercise 9.71. Find the MLE of σ2.
Let z denote a random variable having a normal distribution with μ = 0 and σ...
Let z denote a random variable having a normal
distribution with μ = 0 and σ = 1. Determine each of the following
probabilities. (Round your answers to four decimal places.)
(a)
P(z < 0.20) =
(b)
P(z < −0.20) =
(c)
P(0.30 < z < 0.86) =
Problem 1: confidence interval for a variance parameter for a normal distribution Let Ybe a normal...
Problem 1: confidence interval for a variance parameter for a normal distribution Let Ybe a normal random variable with mean μand variance σ2. Assume that μis known but σ2is unknown. Show that ((Y-μ)/σ)2is a pivotal quantity. Use this pivotal quantity to derive a 1-α confidence interval for σ2. (The answer should be left in terms of critical values for the appropriate distribution.)
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information co...
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale.
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
value at risk at level p, p is between 0 and 1
Q2. Denote by Ф the standard normal distribution (that is, with mean zero and variance 1), and by ф_l the corresponding quantile function. Let X be a normal random variable with mean μ and variance σ2. Evaluate VaRx (p) in terms of ф-1, , 1 and σ. Hint: X = ơZ + μ, where Z is standard normal
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...
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 >...
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...
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and variance σ , with known μ and unknown σ' . You are given that Σ(X-μ)2 is sufficient for σ a) Find El Σ(X-μ). |. Show all steps. Use the fact that: Var(Y)-E(P)-(BY)' i-1 b) Find the MVUE of σ.
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased estimator for the common standard deviation of two independent sam ples Let Yi, Yi2, ..., Yini denote the random sample of size n from the first population with population mean μ| and population variance σ, and let Y21, Y22, ..., Y2na denote an independent random sample of size n2 from the second population with population mean μ2 and population mean ơ3. Sup- pose that...
QUESTION：
Yi, Y2, Y, denote a random sample from the normal distribution with known mean μ 0 and unknown variance σ 2, find t 1 he method-of-moments estimator of σ 2 C2. Continue with Exercise 9.71. Find the MLE of σ2.
Let z denote a random variable having a normal
distribution with μ = 0 and σ = 1. Determine each of the following
probabilities. (Round your answers to four decimal places.)
(a)
P(z < 0.20) =
(b)
P(z < −0.20) =
(c)
P(0.30 < z < 0.86) =
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale.
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
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