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Problem 2 [17 points]. Transformations! a) (5 points) Suppose the time, W, it takes to complete...
(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...
-wa exp{-(20 )2}, where The Normal(μ,02) distribution has density f(x) -oo < μ < oo and...
-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.
8. Suppose W and Z have a bivariate normal distribution 1 1 2(1-p2) (-2pzw+u2) fzw (z,...
8. Suppose W and Z have a bivariate normal distribution 1 1 2(1-p2) (-2pzw+u2) fzw (z, w) 27T 1 (i) Find the marginal density of W then compute its MGF Mw and use it to find the mean and the variance of W. [3 (ii Find fzw (z|w) and use it to identify the distribution of Z given W = w aW bwhere a, b E R. [2 (iii) Derive the density of Y (iv) Compute the mean and variance...
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...
Suppose that a random variable is normally distributed with mean μ and variance σ2 and we...
Suppose that a random variable is normally distributed with mean μ and variance σ2 and we draw a random sample of 5 observations from this distribution. What is the joint probability density function of the sample?
Please help with Q3,4,5 and provide steps. Will rate. Q3 (2 points): Suppose that X takes...
Please help with Q3,4,5 and
provide steps. Will rate.
Q3 (2 points): Suppose that X takes values between 0 and 1 and has probability density function f(x) = 2x. Compute Var(x) and Var(x2). Hint 1: compute E(X), E(X2), and E(X4). Hint 2: Var(Y) = E(Y2) – (E(Y))2 Q4 (2 points): For a random variable X that follows a normal distribution with mean 3 and variance 4: N(u =3, 02=4) What is the probability of X>4? What is the probability of...
find mean and variance ,MGF of one random variable derive that step by step for number...
find mean and variance ,MGF of one random variable
derive that step by step for number 2,3,4.Thank you
2 Chi-square f(x)= 22)/72 2 Exponential Gamma 0<α M (t) = (1-et)" t < Normal N (μ, σ2) E (X) = μ, Var(X) = σ2
Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the dis...
Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the distribution. (1 point). Hint: use the fact that PDF of a density always integrates to 1. (b) Show that the mean of the distribution is u (proof needed). (1 point) (c) Using random sample X1, ,Xn to derive the maximum likelihood estimator of μ (2...
. Suppose that Y is a normal random variable with mean µ = 3 and variance...
. Suppose that Y is a normal random variable with mean
µ = 3 and variance σ
2 = 1; i.e.,
Y
dist = N(3, 1). Also suppose that X is a binomial random variable
with n = 2 and p = 1/4; i.e.,
X
dist = Bin(2, 1/4). Suppose X and Y are independent random
variables. Find the expected
value of Y
X. Hint: Consider conditioning on the events {X = j} for j = 0, 1,
2.
8....
3.1 There is a random variable X with observations {X1,X2, ..., Xn). It is known that...
3.1 There is a random variable X with observations {X1,X2, ..., Xn). It is known that these observations follow the normal distribution with mean μ and variance σ2. Which of the following will lead to a standard normal distribution? (a) (X-A)/o (b) (X- )/a2 (c) (X + μ)/o2 (d) (X + μ)/σ 3.2 In standard normal distribution, 99.7% of observations lie in the range between 3.3 A cumulative distribution function of a random variable Xis by definition a probability that...
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...
-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.
8. Suppose W and Z have a bivariate normal distribution 1 1 2(1-p2) (-2pzw+u2) fzw (z, w) 27T 1 (i) Find the marginal density of W then compute its MGF Mw and use it to find the mean and the variance of W. [3 (ii Find fzw (z|w) and use it to identify the distribution of Z given W = w aW bwhere a, b E R. [2 (iii) Derive the density of Y (iv) Compute the mean and variance...
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 help with Q3,4,5 and
provide steps. Will rate.
Q3 (2 points): Suppose that X takes values between 0 and 1 and has probability density function f(x) = 2x. Compute Var(x) and Var(x2). Hint 1: compute E(X), E(X2), and E(X4). Hint 2: Var(Y) = E(Y2) – (E(Y))2 Q4 (2 points): For a random variable X that follows a normal distribution with mean 3 and variance 4: N(u =3, 02=4) What is the probability of X>4? What is the probability of...
find mean and variance ,MGF of one random variable
derive that step by step for number 2,3,4.Thank you
2 Chi-square f(x)= 22)/72 2 Exponential Gamma 0<α M (t) = (1-et)" t < Normal N (μ, σ2) E (X) = μ, Var(X) = σ2
Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the distribution. (1 point). Hint: use the fact that PDF of a density always integrates to 1. (b) Show that the mean of the distribution is u (proof needed). (1 point) (c) Using random sample X1, ,Xn to derive the maximum likelihood estimator of μ (2...
. Suppose that Y is a normal random variable with mean
µ = 3 and variance σ
2 = 1; i.e.,
Y
dist = N(3, 1). Also suppose that X is a binomial random variable
with n = 2 and p = 1/4; i.e.,
X
dist = Bin(2, 1/4). Suppose X and Y are independent random
variables. Find the expected
value of Y
X. Hint: Consider conditioning on the events {X = j} for j = 0, 1,
2.
8....
3.1 There is a random variable X with observations {X1,X2, ..., Xn). It is known that these observations follow the normal distribution with mean μ and variance σ2. Which of the following will lead to a standard normal distribution? (a) (X-A)/o (b) (X- )/a2 (c) (X + μ)/o2 (d) (X + μ)/σ 3.2 In standard normal distribution, 99.7% of observations lie in the range between 3.3 A cumulative distribution function of a random variable Xis by definition a probability that...
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