Let X1, X2, X3 form a random sample from U(0,1). Find a- The distribution of the second order statistics. b- Find E X(2)].
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![- Let X1, X2, X3 form a random sample from U(0,1). Find
a- The distribution of the second order statistics.
b- Find E X(2)]-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04d0cd5d-c1c9-4182-96f1-d98c5408e40b%2F8f4f8c3e-9561-4fc6-84a9-ae86c99dc169%2Fj5zizxo_processed.jpeg&w=3840&q=75)
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- Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.Let X be a Poisson random variable with E(X) = 3. Find P(2 < x < 4).
- X is a continuous random variable taking values between 0 and 2. If F(1.5) = 0.70 and P(X<1) = 0.20, what is P (1 < X < 1.5)?Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.
- There are two traffic lights on a commuter's route to and from work. Let X1 be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning from work. Suppose that these two variables are independent, each with the pmf given in the accompanying table (so X1, X2 is a random sample of size n = 2). x1 0 1 2 p(x1) 0.1 0.2 0.7 ? = 1.6, ?2 = 0.44 (a) Determine the pmf of To = X1 + X2. to 0 1 2 3 4 p(to) (b) Calculate ?To. ?To = How does it relate to ?, the population mean? ?To = · ? (c) Calculate ?To2. ?To2 = How does it relate to ?2, the population variance? ?To2 = · ?2Let X1, X2,...Xn be a random sample of size n from a normal distribution with mean u and variance o2. Let Xn denote the sample average, defined in the usual way. PROVE E [Xn] = uX is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2