Suppose that two random variables X and Y have the joint PDFfXY(u, v) = {60(u^2)v                       u ≥ 0, v ≥ 0, and u + v ≤ 1,                   0                                    otherwise.(a) Are X and Y independent?(b) What is the marginal distribution fX(t)?(c) What is Pr[X ≥ Y]? (To set up the right integral, it might help you to draw the rangeof (X, Y) in the uv-plane and identify the region within that range where u ≥ v.)

Let Y > 0 be a continuous random variable representing time from regimen start to bone-marrow transplant. Everyone does not survive long enough to get the transplant. Let X > 0 be a continuous random variable representing time from regimen start to death. We can assume X ⊥ Y and model time to death as X ∼ Exp(rate = θ) and time to transplant as Y ∼ Exp(rate = µ). Where Exp(rate = λ) denotes the exponential distribution with density f(z | λ) = λe−λz for z > 0 and 0 elsewhere - with λ > 0. Find the probability that the patient would die before receiving transplant.

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.

(3) Let X = b >(8,-) find E(5+6x) and distribution function.

(26) Let X b(8,- Var-x) and distribution function.

Suppose X = 1/(1+Y), where 3 Y ~ fy(y) = 12 1+y y > 0, 0, otherwise. Identify the distribution of X by name and parameter values.

Suppose that X follows the Weibull distribution with pdf f(r) = Br1e on r> 0, %3D where B> 0. (a) By integrating using the substitution u = , and checking what happens to the integration limits, show that E(X)-r (등+1). %3D (b) Calculate the numerical value of the variance of the Weibull distribution when B= }.

the pdf of a random variable x is given by f(x;θ) = θx^θ−1 x ∈ (0,1). find the maximum likelihood estimate of θ

Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.