Let X = (X,,X,,..., X„) be a random sample with size n taken from population has probability density function fx (x;0) = e®-*) ,x 2 0. Prove that U = min(X)is a sufficient statistic for 0.
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- Let X and Y be independent Exp(2) random variables. Define W = X + Y. a) Determine the correlation between X and W. The joint pdf of X and W is: fx,w(x, w) = 1^(2) e^(-1w) I{OSuppose that f(x)=k/(4^x), x=1,2,3,... is the probability function for a random variable X. Find the value of 1/k.Let X1, X2, ..., X„ be a random sample from a Uniform(0,0 +1) population with 0 > 0, *(4)"^( +1) be the median of the random sample. X where n is an even integer. Let M Find the probability density function of the median M. Find the expected value and variance of M.Q2) The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function as shown below, Find the probability that a bottle of this medicine will have a shelf life of anywhere between 40 and 120 days 20000 0Let the density function of a random variable X be given by fx(x) = 0x®-1, 0Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)Let the probability function of the random variable X be defined as f(x)=cx^4 for x > 1, otherwise f(x)=0. Calculate the constant c, the expected value and the variance.Suppose that f(x)=2/(3^x), x=1,2,3,... is the probability function for a random variable X. Find P(X>3). Use 4 decimal places.What is the expected value of a continuous random variable X with a probability density function f(x) = (1/sqrt(2 * pi)) * e^-((x^2)/2)?Q2) The shelf life, in days, for 3 نقاط bottles of a certain prescribed medicine is a random variable having the density function as shown below, Find the probability that a bottle of this medicine will have a shelf life of anywhere between 40 and 120 days 20000 0 < x f(x) = }(x + 100)³ otherwise 0.184 O 0.139 O 0.125 0.304 0.238Let X and Y be two independent continuous random variables with the following probability density functions: f(x) = (1/2)e^(-x/2) for x > 0 g(y) = (1/3)e^(-y/3) for y > 0 What is the probability density function of Z = X + Y?Q2) The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the below density function ,find the probability that a bottle of this medicine will have a shelf life of anywhere between 50 and 120 * days 20000 0SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,