Using the law of the unconscious statistician, consider observing random variables U₁₂ (f(x) 1(x[0,1])) and calculate E(g(U)). ~ Unif(0,1)
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![Using the law of the unconscious statistician, consider observing random variables U₁₂
(f(x) 1(x[0,1])) and calculate E(g(U)).
~
Unif(0,1)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1201f5c7-7e92-4f63-af45-4d5288811567%2F87fa5853-b82f-471e-b0c3-4eac2a7ea340%2Fqikpauq_processed.jpeg&w=3840&q=75)
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- Let X be a random variable with p.d.f., fx) = {2e2d-»),x > 1 0 ,x < 1. (1) Find the moment generating function of X (2) Find the mean and variance of X using the mgf from (1).If(x,,y; ).(x,, y; ),..(x,,y«) be a random sample taken from a logistic regression, then the log-likelihood cquation is, a) ,In| Eln(1-z,) -T, b) Σh(1 -π) Σy n -Σh(π) 1-7, c) i-1 1-1 d) Σyn(1- r)+Σ -Q3/A/ If M,(t) = 1-2r ,t 1) i - O.w
- Suppose the random variable X~ Beta(a, 3), namely its PDF fx(x) = I(a + 3) r(a)(3) -ra-1(1−z)3-1, 0 < x < 1, Beta(a +3, y), namely its PDF fy (y) = I(a +8+y) r(a + 3)(y) ¹(1 − y)*-¹, 0 < y < 1, and X and Y are independent. Define U = XY, V = X. Find the joint PDF fuv(u, v).Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of YLet X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.