Below are few properties of a normal
Please prove the following theorem: Let Yı, Y2, ... ,Yn be independent normally distributed random variables...
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.
Question 4 Let Yı: Y2, .... Yn denote a random sample and let E(Y) = u and Var(Y) = o-y, i = 1, 2, ..., n. (b) Prove that the standard error of the sample mean Y SEⓇ) =
1. Let Yı,Y2,..., Yn denote a random sample from a population with mean E (-0,) and variance o2 € (0,0). Let Yn = n- Y. Recall that, by the law of large numbers, Yn is a consistent estimator of . (a) (10 points) Prove that Un="in is a consistent estimator of . (b) (5 points) Prove that Vn = Yn-n is not a consistent estimator of (c) (5 points) Suppose that, for each i, P(Y, - of ? Prove what...
Let Yı, Y2, Ys, and Y4 be independent, identically distributed random variables from a mean u and a variance 02. Consider a different estimator of u: W=Y+Y2+2Y3+ Y 00 This is an example of a weighted average of the Y a) Show that W is a linear estimator. b) Is W an unbiased estimator of u? Show that it is - or it isn't (E(W) = Find the variance of W and compare it to the variance of the sample...
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 Yı, Y2, ..., Yn iid N (4,02), where the population mean and population variance o2 are both unknown. Show that the Method of Moments (MOM) estimators of u and o2 are given by n û =Ý ΣΥ, n i=1 п n - - 1 ô2 - = s2 Ü(Y; – 7) n п i=1 Note: In this case, (Y, S2) is a sufficient statistic for (u, 02). The MOM estimators of u and o2 are therefore functions of a...
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
(1 point) Let Yı, Y2, ..., Yn be a random sample from the probability density function f(yla) = |aya-2/5° f(y ) 0 <y< 5 otherwise 0 for > -1. Find an estimator for a using the method of moments.
Let Yı,Y2, ..., Yn be iid from a population following the shifted exponential distribution with scale parameter B = 1. The pdf of the population distribution is given by fy(y\0) = y-0) = e x I(y > 0). The "shift" @ > 0 is the only unknown parameter. (a) Find L(@ly), the likelihood function of 0. (b) Find a sufficient statistic for 0 using the Factorization Theorem. (Hint: O is bounded above by y(1) min{Y1, 42, ..., .., Yn}.) (c)...
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)