5. A statistical model for the data y = (y1,.. (Y1,, y8) is a random sample from normal distribution N(μ, 1) i.e 1 ƒ (y | μ) = (2m)*/2 xp(-Σ 1 Σ (Yi - µ)²) 2 where μ is a real parameter and variance is known number 1. Prior p(μ) is chosen to be the density of the normal distribution N(0, 22), where the variance is a known positive number 4. Show that the posterior p(µ | y) follows normal distribution N(μ₁, 0). What are the numbers μ₁ and σ?? The simplest way is to apply the proportionality p(µ | y) ∞ p(µ)f(y | µ).

Let X and Y have the joint pdf f(x,y)= x+y , 0<=x<=1, 0<=y<=1. Calculate the mean(x) mean (y) variance (x) variance(y)

Find the characteristic function of the geometric distribution given by P(X = r) = q' p, r = 0, 1, 2, ..., ∞, p + q = 1. %3D Hence find the mean and variance.

Suppose that X ~ and variance 1, i.e., (Y X = x) ~ N(2x, 1). Find the marginal pdf of Y. N(0,1). Then, under the condition X = x for some r E R, the distribution of Y is normal with mean 2x

The random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−3/4. Calculate Var(X+Y) with a full explanation

Suppose X = 1/(1+Y), where 3 Y ~ fy(y) = 12 1+y y > 0, 0, otherwise. Identify the distribution of X by name and parameter values.

Let Y, represent the th normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Y₁, Y2. Yin, of size n₁, from the ith population with sample mean Y, and sample variance S² = ₁₁-1(Y₁-₁². j=1 (g) For non-zero constants a's, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of = 0² = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #₁ and μ2, when o² = 0 = 0² is known. (j) Define appropriate rejection regions, in terms of Y₁ - Y2, for testing Ho: #₁ = 2 against a two-sided alternative hypothesis at the a level of significance.

Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2 is known. Find the Score function, the estimating equation and the information matrix.

Let X ~ U[0,1] and Y = -βln(1-X). What is the distribution of Y? Justify.

2. Find the normal equations to the curve 2* = ax + bx + c %3D 2 = ax+ bx + c