) Let Y ∼ Exp(λ). Given that Y = m, let X ∼ Pois(m).
Find the mean and
variance of X.
2.
X | Y = m ~ Poisson(m) => E[X | Y = m] = m and Var[X | Y = m] = m
By law of Total Expectation,
E[X] = E[E[X | Y = m]] = E[m]
m = Y ~
Exp(
)
By law of Total Variance,
Var(X)=E[Var(X|Y = m)]+Var(E[X|Y = m])
= E[m] + Var[m]
m = Y ~ Exp(
)
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2. (Anderson, 10.11) Let Y ~Exp(A). Given that Ym, let X ~Pois(m). Find the mean and variance of X.
Suppose X~Pois(A) and Y ~Pois(2A) are independent random variables. Consider a linear estimator of λ, that is, λ = aX + bY. (a) Find an expression for the bias of λ, in terms of a and b, and determine a condition on the values of a and b, such that λ is unbiased. (b) Of all the values of a and b that make the estimator unbiased, find the values of oa and b that minimize the variance of the...
Let X and Y be independent rv’s with pmf Pois(λ1) and Pois(λ2),
respectively.
(a) Find the distribution of Z = X + Y .
(b) Find the distribution of X|X + Y .
(c)If X∼Pois(λ1) and Y|X=x∼Bin(x,p). Find the distribution of
Y.
Let X and Y be independent rv's with pmf Pois(11) and Pois(12), respectively. (a) Find the distribution of Z= X+Y. (b) Find the distribution of X|X +Y. (c) If X ~ Pois(11) and Y|X = x ~ Bin(x,p)....
Let X and Y be independent rv's with pmf Pois(11) and Pois(12), respectively. (a) Find the distribution of Z = X+Y. (b) Find the distribution of X X +Y. (c) If X Pois(11) and Y|X = r ~ Bin(x,p). Find the distribution of Y.
Suppose X~Pois(λ1) and Y~Pois(λ2). Find the conditional mass function for X given X+Y = m
2. (30 pts) Let X and Y be independent rv's with pmf Pois(41) and Pois(12), respectively. (a) Find the distribution of Z = X +Y. (b) Find the distribution of X X +Y. (c) If X ~ Pois(11) and Y|X = x ~ Bin(x,p). Find the distribution of Y.
2. (30 pts) Let X and Y be independent rv's with pmf Pois(11) and Pois(2), respectively. (a) Find the distribution of Z = X+Y. (b) Find the distribution of X|X+Y. (c) If X Pois (41) and Y|X = x~ Bin(x, p). Find the distribution of Y.
2. (30 pts) Let X and Y be independent rv's with pmf Pois(41) and Pois(12), respectively. (a) Find the distribution of Z = X+Y. (b) Find the distribution of X|X +Y.. (c) If X ~ Pois(11) and Y|X = x ~ Bin(x,p). Find the distribution of Y.
X~exp(λ) with λ=1 1) define Y= X^1/2. Find the support of Y and its density. 2) define Z = X^2 + 2X. Find the support of Z and its density.
(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼ Poisson(λ + γ). (b)Let X1, . . . , Xn be an iid random sample from Poisson(λ), provide a sufficient statistic for λ and justify your answer. (c)Under the setting of part (b), show λb = 1 n Pn i=1 Xi is consistent estimator of λ. (d)Use the Central Limit Theorem to find an asymptotic normal distribution for λb defined in part (c), justify...