Suppose X ~ Poisson(2λ) & Y ~ Poisson(3λ) are independent. Show that T = (.32)X +...

Question

Question

Suppose X ~ Poisson(2λ) & Y ~ Poisson(3λ) are independent.

Show that T = (.32)X + (.12)Y is an unbiased estimator of λ & determine Var(T).

Hint: begin by computing E(T).

math Statistics-And-Probability

Answer 1

Answer #1

we hou Ahckf 0.322A) +12(3 晰)

Answer 2

Similar Homework Help Questions

(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼...

(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...
Suppose X~Pois(A) and Y ~Pois(2A) are independent random variables. Consider a linear estimator of λ, that...

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...
Suppose X1,. , Xn are iid Poisson(A) random variables. Show by direct calculation without using any...

Suppose X1,. , Xn are iid Poisson(A) random variables. Show by direct calculation without using any theoremm in mathematical statistics, that (a) Ση! Xi/n is an unbiased estimator for λ. (b) X is optimal in MSE among all unbiased estimators. This is to say, let T be another unbiased estimator, then EA(X) EA(T2

1. Let {x, t,f 0) and {Yǐ.12 0) be independent Poisson processes,with rates λ and 2A, respectivel...

1. Let {x, t,f 0) and {Yǐ.12 0) be independent Poisson processes,with rates λ and 2A, respectively. Obtain the conditionafdistributiono) Moreover, find EX Y X2t t given Yt-n, n = 1,2. 2, (a) Let T be an exponential random variable with parameter θ. For 12 0, compute (b) When Amelia walks from home to work, she has to cross the street at a certain point. Amelia needs a gap of a (units of time) in the traffic to cross the...
5. Suppose X ~ Poisson(A = 5) and Y ~ Poisson(λ = 10), and they are...

5. Suppose X ~ Poisson(A = 5) and Y ~ Poisson(λ = 10), and they are independent. Using the moment generating function method, find the distribution of Z = X + Y.
X and Y are random variables (a) Show that E(X)=E(B(X|Y)). (b) If P((X x, Y )...

X and Y are random variables (a) Show that E(X)=E(B(X|Y)). (b) If P((X x, Y ) P((X x})P({Y y)) then show that E(XY) = E(X)E(Y), i.e. if two random variables are independent, then show that they are uncorrelated. Is the reverse true? Prove or disprove (c) The moment generating function of a random variable Z is defined as ΨΖφ : Eez) Now if X and Y are independent random variables then show that Also, if ΨΧ(t)-(λ- (d) Show the conditional...

Problem 3. Let X1, . . . , Xn be independent Poisson(λ) random variables. Find a...

Problem 3. Let X1, . . . , Xn be independent Poisson(λ) random variables. Find a BUE of e−2λ . Hint: Compute Pλ(X1 = 0, X2 = 0)
4. Let ,, , xn be independent and suppose that E(X.) k,0 + bi, for known...

4. Let ,, , xn be independent and suppose that E(X.) k,0 + bi, for known constants ki and bi, and Var(X) = σ2, i 1, , n. (a) Find the least squares estimator θ of θ. (b) Show that θ is unbiased. c) Show that the variance of θ is Var(8)-: T (e) Show that the variance of is Var() (d) Show that Tn Σ(x,-ke-W2 = Σ(x,-k9-b)2 + Σ ka@ー0)2 i-1 -1 ー1 (e) Hence show that Ti 121
4. Given a Poisson process X(t), t > 0, of rate λ > 0, let us fix a time, say t-2, and let us con...

4. Given a Poisson process X(t), t > 0, of rate λ > 0, let us fix a time, say t-2, and let us consider the first point of X to occur after time 2. Call this time W, so that W mint 2 X() X(2) Show that the random variable W - 2 has the exponential distribution with parameter A. Hint: Begin by computing PrW -2>] for 4. Given a Poisson process X(t), t > 0, of rate λ...

Suppose that X and Y are independent random variables with the same unknown mean u. Both...

Suppose that X and Y are independent random variables with the same unknown mean u. Both X and Y have a variance of 36. Let T = aX + bY be an estimator of u. What condition must a and b satisfy in order that T be an unbiased estimator for ? Is T a normal random variable?