闪电代写 -代写CS作业_CS代写_Finance代写_Economic代写_Statistics代写_代码代做_IT代写_加急帮助

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

STATS 100B Homework 2

2022

Problem 1 (30 pts)

If X is a nonnegative integer-valued random variable, the probability-generating function of X is

deﬁned to be

G(s) = ! skpk

k=0

where pk = P(X = k).

a. Show that

pk = G(s)"

(5 pts)

dG " = E[X]

d2G " = E[X(X − 1)]

(10 pts)

c. Express the probability-generating function in terms of moment-generating function. (5 pts)

d. Find the probability-generating function of the Poisson distribution. (10 pts)

Problem 2 (10 pts)

Show how to ﬁnd E(XY) from the joint moment-generating function of X and Y .

Hint: use the Taylor expansion of a real function with two variables, which we covered in the lecture and is also available at . . T S .http://mathworldwolframcom/aylorerieshtml

Problem 3 (15 pts)

Use moment-generating functions to show that if X and Y are independent, then Var(aX + bY) = a2Var(X) + b2Var(Y)

Hint: Var(X) = E[X2] − (E[X])2 = M(0) − M(0)2; MX+Y (t) = MX (t)MY (t) if X and Y are independent.

Problem 4 (15 pts)

Use the delta method to ﬁnd the approximate expressions for the mean and variance of Y = g(X) in terms of µX = E[X] and σ = Var[X] for

a. g(x) = √x (5 pts)

b. g(x) = log(x) (5 pts)

c. g(x) = sin− 1 (x) (5 pts)

Problem 5 (30 pts)

The position of an aircraft relative to an observer on the ground is estimated by measuring its distance r from the observer and the angle θ that the line of sight from the observer to the aircraft makes with the horizontal. Suppose that the measurements, denoted by R and Θ , are subject to random errors and are independent of each other. Suppose Var[R] = σ , and Var[Θ] = σ . The altitude of the aircraft is then estimated to be Y = R sinΘ .

Note that (r, θ) are considered as realizations of (R, Θ) and will be treated as numbers. Here we do not know E[R] and E[Θ].

a. Find an approximate expression for the mean of Y in terms of r , θ , σ and σ . (10 pts)

Hint: use the delta method for the bivariate case; use Taylor expansion at (r, θ); for this problem only, assume E[R] ≈ r and E[Θ] ≈ θ .

b. Find an approximate expression for the variance of Y in terms of r , θ , σ and σ . (10 pts) Hint: use the delta method for the bivariate case; use Taylor expansion at (r, θ).