Let Y obey a normal distribution with the parameter (µ, σ2), and assume that the conditional distribution of X is a normal distribution with the parameter (y, 1) under the condition Y = y. a. Find EX, Var(X), Corr(X, Y) b. Prove that (X, Y) union obeys normal distribution c. Find E[Y|X = x].
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Let Y obey a
a. Find EX, Var(X), Corr(X, Y)
b. Prove that (X, Y) union obeys normal distribution
c. Find E[Y|X = x].
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- Let X - U[0, 1]. Find the distribution of Y = 1 - X.Let X and Y be independent standard normal random variables. Determine the pdf of W = x² + y². Find the mean and the variance of U = /W. Let Y₁, Y₂, ..., Yn denote a random sample of size n from a population with a uniform distribution on the interval (0, 0). Consider = Y(1) = min(Y₁, Y₂, ..., Y₁) as an estimator for 0. Show that is a biased estimator for 0. Let X and Y be independent exponentially distributed random variables with parameter X λ = 1. If U = X + Y and V =. Find and identify the marginal density of U. X+YLet Y₁,..., Yn be an iid sample U(0, 0), 0 > 0, with pdf { 1/0, 0≤ y ≤0, 0, elsewhere and CDF f(y) = F(y) = = 0, y < 0, y/0, 0≤ y ≤ 0, 1, y ≥ 0
- Let X~ N(0, 1), and let Z 1/2) be independent of X. Let Y Y are uncorrelated. Are X and Y independent? = Unif{−1, 1} (i.e. P(Z = −1) = P(Z 1) ZX. What is the distribution of Y? Show that X and = =Let X and Y be independent v.a with uniform distribution on the interval (0, 20) and (0, 30) respectively. Find the function f_X+Y (z).Let X be a random variable normally distributed with mean μx 1. Find the mean μy of Y. 2. Find the standard deviation oy of Y. 3. Find the PDF fy of Y and evaluate fy (5). (My, oy, fy (5)) = ___________). = 6 and standard deviation ox = 4. Let Y = 4X + 5.
- Let X be a random variable with uniform distribution on the interval [-2,2]. Let Y be defined as Y = X5. Calculate the pdf of Y.Find the equation of the normal line to the curve y = 2x - x< at (2, 0). O a. 2x - y = 4 O b.x- 2y = 2 O C. X + 2y = 2 O d. 2x + y = 4Assume that y = [Y₁ _Y2_Y3 Y4] [Y₁ 92 93 94] follows a multi-dimensional normal distribution №₁(μ, Σ) where fl = 2 5 [-₁] Σ 8 0 0 3 -1 0 020 ROL T -1 0 5 20 Use properties of multi-dimensional normal distributions to an- swer the following: i. Find the distribution of y₁ and the distribution of y3; ii. What is the joint distribution of y₁ and y₂? iii. Compute p(Y3|Y4 = 1). iv. Determine the random variables that are independent. v. Find the distribution of 2 = = 4y₁ − 2y2 + Y3 − 3y4.