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- 2.) Use the moment generating function you just proved to verify that for a Poisson distribution, E(X) = λ and Var(X) = λ.Let X have a Poisson distribution. If , find the mode of the distribution. Suppose Y has a uniform pdf over the interval [3,5]. Find its (1) first quartile, its (2) median, and (3) the sum of its third quartile and its ninth decile.area to right of 2.821 under a t-distribution with 9 degrees of freedom
- If X has an F distribution with ν1 and ν2 degrees offreedom, show that Y = 1Xhas an F distribution with ν2 and ν1 degrees of freedom.Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.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)?
- 1. Let X have a gamma distribution with α > 1. Show thatE [1/X] = 1/[θ*(α −1)]If N = 48, find tcv and use it to draw a t distribution with the rare and common zones labeled. Assume the tes is 2 tailed and alpha is set at .05Suppose X1, . . . , Xn are i.i.d. from a Poisson distribution with parameter λ, which is unknown. (a) Find with justification the MLE for λ.(b) Find the standard error of your estimate in part (a). (c) Can you estimate the standard error from part (b)?
- Suppose the distribution of the time $X$ (in hours) spent by students at a certain university on a particular project is gamma with parameters $\alpha=50$ and $\beta=2 .$ Because $\alpha$ is large, it can be shown that $X$ has approximately a normal distribution. Use this fact to compute the approximate probability that a randomly selected student spends at most 125 hours on the project.The lifetime X (in 100’s of hours) of a certain type of vacuum tube has a Weibull distribution with parameters α = 0.25, and β = 3. What is P (X ≥ 75)?Suppose X has a uniform distribution on the interval [−2, 3]. Find themoment generating function of X.