Use the method of moment to estimate the parameter 0 for the density f (x; 0) =; I[0,0](x) based on a random sample X1,X2,...,Xn. Give a sufficient statistic for 0.
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based on a random sample X1, X2,...,Xn. Give a sufficient statistic for 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bf7f571-1f5c-4f23-a40a-002fd0e5042d%2F52d75757-5461-4413-a491-021a6551bc42%2Fs0swa0g_processed.png&w=3840&q=75)
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- Use the rejection method to generate a random variable having density function f(x) = kx1⁄2e−x , x > 0 where k =1/Γ (3/2) =2√πFind the moment generating function for the random variable X whose density function is f(x). fx = 2x 0<x<1 MX(t) = _______________________________________1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.
- a. What is the joint marginal PDF of X and Z?b. What is [P 0 <= Y <= 3/2 | X = 1/3 Z = 5/2]?Consider the random variable X with PDFf(x) = e−x / (1 + e−x)2 , x ∈ R.Find the density function of Y = 1/ (1 + e−X)For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
- If X and Y are independent exponential random variables, each having parameter λ.(a) Find the joint density function of U = X + Y by using the convolution of fX and fY .(b) Find the joint density function of V = X − Y by using the method of transformation.(c) Are U and V independent?Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.Let X be a continuous random variable with density function f(x) = 2x, 0 ≤ x ≤ 1. Find the moment-generating function of X, M(t), and verify that E(X) = M′(0) and that E(X2) = M′′(0).
- 1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.2. Identify the probability density function, then find the mean and variance without integrating. b. f(x) =1/6 e^−x/6, [0,∞) c. f(x) =1 / 3√2π e^−(x−16)^2/18, (−∞,∞)Let x and y be joint continuous random variable with joint pdf f XY (x, y) = {?x + 1 , x, y ≥ 0, x + y < 10, ??ℎ??????1. Find the constant c.2. Find the marginal PDF’S fX (x) and fY (y)3. Find P(Y<2X2)