Let X₁, X₂, X, be a random sample from a binomial distribution X with parameter 20 and 0.3, i.e., X~Bin(20,0.3), compute E(X), V(X), E(S²). www
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- Let X ~ Binomial(3,0.5) and Y ~ Poisson(2) The two random variables are independent. Calculate the quantity, Var(2X-5Y)Let Y~Po(1) be a poisson distributed variable with parameter 1 = 1. Suppose that, designated on Y, X is X, i = 1, ...,7 independent random variable with conditional distribution P(X;|Y) = Bin(N, 1/3) Determine the variance of the sum X = X1+...+X,Suppose that Y, Y2,.-.,Y, are independent random variables from a gamma distribution of the parameter (2,6+4). a) Use the method of moment to obtain estimator of B. b) Show that the estimator B, obtained in part a) is unbiased. c) Show that the estimator B is a minimum variance unbiased estimator of ß.
- Suppose Y1,Y2,··· ,Yn are i.i.d. continuous uniform(0,1) distributed.(a) Prove that the kth-order statistic, Y(k), has Beta distribution with α = k and β = n−k+ 1.(b) What is the distribution of the median of Y1,Y2,··· ,Yn when n is an odd integer?(c) Find the joint density of the middle two of Y(1),Y(2),··· ,Y(n) when n is an even integerLet X(1),.., X(n) be the order statistics from Unif(0, 0). Obtain the joint distribution of (, X(m)) and show that they are independent. What is the distribution of ? X(m)Let Y1, Y2,...,Yn be independent random variables with Y; Poisson(A;). Show that the sum of these random variables S has a Poisson distribution with rate paramcter A = di.
- Prove the followingIT HAS BEEN FOUND THAT A FLUCTUATING ELECTRIC CURRENT X AT ONE OF THE ESKOM SUBSTATIONS HAS A UNIFORM PROBABILITY DISTRIBUTION, f(x) = a≤x≤b otherwise a+b (b-a)² WITH E(X) = AND Var (X) = 12 AS THE STATISTICIAN AT ESKOM AND HAVE BEEN GIVEN A RANDOM SAMPLE OF 7 MEASUREMENTS X₁, X₂,,X, OF THE ELECTRIC CURRENT TO INFER ABOUT THE MEAN FLUCTUATION OVER A SPECIFIC PERIOD. AFTER CAREFUL CONSIDERATION YOU HAVE IDENTIFIED THREE POSSIBLE ESTIMATORS 0₁ = X₁+X7, 0₂ = X₁ +2X₂+X AND 3 = X 3 Show which of the estimators possess this propertyTwo random variables X = (1,2) and Y= (-2,-1] are equiprobable and independent. Determine the correlation factor r(X,Y),
- Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.If a discrete random variable X has the following probability distribution: X -2, - 1, 0, 1, 2 P(X) 0.2, 0.3, 0.15, 0.2, 0.15 Use this to find the following: (a) The mean of X and E[X^2]. (b) The probability distribution for Y = 2X^2 + 2 (i.e, all values of Y and P(Y )). (c) Using part (b) (i.e, the probability distribution forY ), find E[Y ]. (d) Using part (a), verify your answer in part (c) for E[Y ]. **Note: Please do not just copy from Chegg!Let X1, X2,...X, be a random sample from f(x,e) = -.0 < x< 8, then T= max(X,} is a %3D complete statistic for the parameter 0 Select one: O True False