A random variable X has the exponential distribution f(x) = hehx₂ x > 0 1. Show that the moment-generating function for t < λ f X is MX (1) = (1 - 1) . Find the mean and variance of X.

Q: Suppose you are given the moment generating function M(t) =- for t< In 3 4-3e Find the mean and…

A: The first derivative of moment generating function is, The mean of the distribution will be: Thus,…

Q: 3. Suppose that X is a continuous random variable with pdf 32², 0<x< 1 0, otherwise Let Y= 2X + 3.…

A: f(x)=3x2; 0&lt;x&lt;1

Q: Let X be a continuous random variable with PDF 2x 0 < x < 1 fx(x) = {0 otherwise Find the expected…

A:

Q: If the PDF of X is f(x)=2x/k2 for 0<x<k, for what value of k is the variance of X equal to 2?

A: The probability density function of Xf(x)=2x/k2      for 0&lt;x&lt;kfor what value of k the…

Q: Find the moment generating function ME(t) for an exponential random variable with parameter (lambda)…

A:

Q: Suppose that f (x) = 0.125x for 0 < x < 4 %3D Determine the mean and the variance of X.

A:

Q: 68. Let X be a continuous random variable with pdf (a) Find the pdf for Y = √X. (b) Find the pdf for…

A: Pdf of X is fx(x)=3x2 ,0,&lt;x&lt;1

Q: Question is attached.

A: Given thatThe moment generating function of a random variable isTo find mean and variance:

Q: If random variable x is exponentially-distributed with mean a= 2, calculate p(X> 3)· Σ 2e-2x x=3 E…

A: From the provided information, Mean (λ) = 2 X~exp(2)

Q: If 1. If for >0 the energy of a pdf is E(x)=r, find the variance. (A) 1 (B) 2 (C) 1/2 (D) Undefined…

A: Step 1:Find the variance, The energy of a pdf is E(x) =x and x&gt; 0.Step 2:The variance is…

Q: Let X1,, X, be a random sample from a uniform distribution on the interval (0, 50), where 0> 0.…

A: Disclaimer: As the question contains multiple parts, only first three will be answered. X1, X2,…

Q: IF it was XG (23) Independent of Xen G (13) Find Miley D-Probability density Function of the…

A: Sour, The prompt states that X1 follows a Gamma distribution with a shape parameter of 2/3, and X2…

Q: The X be a continuous random variable with cumulative distribution function 1 F(x) = 1 for x > 0…

A: From the given information, X be continuous random variable with cumulative distribution is,…

Q: Use the moment-generating function of gamma distribution to show that E(X)=a0 and Var(X)=a®² ||

A:

Q: X is a continuous random variable and the pdf of X is x > 1 f(x) : xs1 Determine the value of k for…

A: Probability density Function -        fxx of the random variable X is define as :-…

Q: 4. Let X₁ and X₂ be independent identically distributed random variables both having exponential…

A: Let X1 and X2 be independent and identically distributed random variables both having an exponential…

Q: X1,X2,...,Xn are i.i.d. random variables from N(0,θ) distribution. Here, θ= σ2 >0 is the variance of…

A:

Q: The moment generating function can be used to find the mean and variance of the normal distribution.…

A: