CS代考程序代写 University of California, Los Angeles Department of Statistics

University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou
Practice 1
Find the distribution of the random variable X for each of the following moment-generating func- tions:
a. MX(t)=􏰉1et+2􏰊5. 33
b.M(t)=et . X 2−et
c. MX(t) = e2(et−1). EXERCISE 2
Let MX(t) = 1et + 2e2t + 3e3t be the moment-generating function of a random variable X. 666
a. Find E(X). b. Find var(X).
c. Find the distribution of X.
EXERCISE 3
Let X follow the Poisson probability distribution with parameter λ. Its moment-generating function is MX(t) = eλ(et−1).
√
a. Show that the moment-generating function of Z = X−λ is given by:
Statistics 100B
EXERCISE 1
√√
MZ (t) = e− λteλ(e λ −1).
b. Use the series expansion of
t
λ
t t (t)2 (t)3 e√λ=1+λ+ λ + λ +···
√√√
to show that
lim M (t)=e1t2. λ→∞Z 2
1! 2! 3!
√
In other words, as λ → ∞, the ratio Z = X−λ converges to the standard normal distribution.
λ
EXERCISE 4
Use the result of part (b) of the previous exercise:
In the interest of pollution control an experimenter wants to count the number of bacteria per small volume of water. Let X denote the bacteria count per cubic centimeter of water, and assume that X follows the Poisson distribution with parameter λ = 100. If the allowable pollution in a water supply is a count of 110 bacteria per cubic centimeter, approximate the probability that X will be at most 110.

EXERCISE 5
Let X1,X2,···,Xn be i.i.d. random sample from N(μ,σ). Using moment genearating functions show that the sum of these n observations T = 􏰃ni=1 Xi also follows the normal distribution. What is the mean and standard deviation of T ?
EXERCISE 6
Suppose that X1,···,Xm and Y1,···,Yn are two samples, with X ∼ N(μ1,σ1) and Y ∼ N(μ2,σ2). The difference between the sample means, X ̄ − Y ̄ , is then a linear combination of m + n normal random variables.
a. FindE(X ̄−Y ̄).
b. FindVar(X ̄−Y ̄).
c. Use moment generating functions to show that the distribution of X ̄ − Y ̄ is normal with mean and variance equal to your answers in (a) and (b).
d. Suppose σ12 = 2, σ2 = 2.5, and m = n. Find the sample sizes so that X ̄ − Y ̄ will be within one unit of μ1 − μ2 with probability 0.95.
EXERCISE 7
If the random variable X follows the normal distribution with μ = 0, σ2 = 1 and Y = eX find the probability density of Y . This is called the lognormal distribution.
EXERCISE 8
If the radius of a circle X is an exponential random variable with parameter λ, find the probability density function of its area Y .