

8.5 Random variables Y1,... , Yn have a joint normal distribution with mean 0 if there exist inde...
Let Y1, Y2, . . . , Yn be independent random variables with Exponential distribution with mean β. Let Y(n) = max(Y1,Y2,...,Yn) and Y(1) = min(Y1,Y2,...,Yn). Find the probability P(Y(1) > y1,Y(n) < yn).
Let the independent normal random variables Y1,Y2, . . . ,Yn have the respective distributions N(μ, γ 2x2i ), i = 1, 2, . . . , n, where x1, x2, . . . , xn are known but not all the same and no one of which is equal to zero. Find the maximum likelihood estimators for μ and γ 2.
Let Y1, Y2, ..., Yn be independent random variables
each having uniform distribution on the interval (0, θ).
(a) Find the distribution of Y(n) and find its expected
value.
(b) Find the joint density function of Y(i) and Y(j) where 1 ≤ i
< j ≤ n. Hence
find Cov(Y(i)
, Y(j)).
(c) Find var(Y(j) − Y(i)).
Let Yİ, Ya, , Yn be independent random variables each having uniform distribu- tion on the interval (0, 6) (a) Find the distribution...
Let Y1, Y2, ..., Yn be independent random variables
each having uniform distribution on the interval (0, θ)
(c) Find var(Y(j) − Y(i)).
Let Y İ, Y2, , Yn be independent random variables each having uniform distribu- tion on the interval (0,0) Let Y İ, Y2, , Yn be independent random variables each having uniform distribu- tion on the interval (0,0)
Let Y1, Y2, ..., Yn be independent random variables each having
uniform distribution on the interval (0, θ).
Find variance(Y(j) − Y(i))
Let Yİ,Y2, , Yn be independent random variables each having uniform distribu - tion on the interval (0,0) Fin ar(Y)-Yo
8.7-11. Let Y1,Y2, ...,Yn be n independent random variables with normal distributions N(Bx;,02), where X],x2,...,xn are known and not all equal and B and 2 are unknown parameters (a) Find the likelihood ratio test for Ho: B = 0 against H: B+0. (b) Can this test be based on a statistic with a well-known distribution?
Assume Y1, ...,Yn are IID normal random variables where mean μ and variance "2 are both unknown. Assume that ¯ Y = 0, and s, the sample standard deviation, equals sqrt(n). Compute a 1 − a confidence interval for the mean μ. Leave your answer in terms of ta/2, the critical value for a t distribution. How many degrees of freedom does this t distribution have?
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(H = 0) = q = 1-p. (Such random variables are called Bernoulli random variables.) a Find the moment-generating function for the Bernoulli random variable Y b Find the moment-generating function for W = Yit Ye+ … + . c What is the distribution of W? 1.
Let Xi, X2, , xn be independent Normal(μ, σ*) random variables. Let Yn = n Ση1Xi denote a sequence of random variables (a) Find E(%) and Var(%) for all n in terms of μ and σ2. (b) Find the PDF for Yn for all n c) Find the MGF for Y for all n
Suppose Y1, Y2, …, Yn are independent and identically distributed random variables from a uniform distribution on [0,k]. a. Determine the density of Y(n) = max(Y1, Y2, …, Yn). b. Compute the bias of the estimator k = Y(n) for estimating k.