- Lei x1, X2,.n be a random sample from N(u, o?), calculate the cstimators of the popuiation parameters using the moment generating function technique.
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- Use the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.Let Mx, y be the moment generating function of random variables that are not independent of X and Y. Which of the following / which are not the properties of the function Mx, y?Use the moment generating function to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n.Find the distribution of Y = X1 + · · · + Xn.
- Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50Find kurtosis by using ungroup data given in the picture using quartile deviation, make sure to arrange data before solving question.If X, Y are normally distributed independent random variables, then show that W = 2X - Y is normally distributed. Do not use moment generating function, only use convolution formula.
- Find the moment generating function of the continuous random variable X∼U (a, b).The probability that the Air Conditioning of a brand new car is defective is 10%. Let X be the number of cars with defective AC in a sample of size 100. (a). Find P(X < 6) exactly. (b). ) Find P(X < 6) approximately, using a Poisson approximation (c). Find P(X < 6) approximately, using a normal approximation. Note: For parts (a) and (b), you only need to provide a formal expression each. For part (c), you need to compute the numerical values using the Q-function table (Table 4.2) from the textbook.Let X~N(0,1). Let Y=2X. Find the distribution of Y using the moment generating function technique.
- Assume that the variables Y1, Y2,... in a compound Poisson process have Bernoulli distribution with parameter p . Show that the process reduces to the Poisson process of parameter λp.Let X1, X2, ... Xn random variables be independent random variables with a Poisson distribution whose parameters are l1, l2, ... ln, respectively. Which of the following is the moment generating function of the random variable Z defined as (the little image)?Choose a point at random in (0,1), then this point divides the interval (0,1) into two subintervals. What is the expected length of the subinterval covering a given point s with 0 < s < 1?