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- Suppose that X₁, X2, X3 are independent and identically distributed random variables with distribution function: Fx (x) = 1 – 3¯ª for x ≥ 0 and Fx (x) = 0 for x 1).Suppose that X₁, X2, X3 are independent and identically distributed random variables with distribution function: Fx (x)=12* for x ≥ 0 and Fx (x) = 0 for x 4).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 Y
- Let X1, X2, X3 be independent Exp()-distributed random vairables. Determine the pdf of X(1)X(3)Let X1, X2,..., Xn be a random sample from a population with a pdf a+2022 (a+ 2021) („) ;x > 0, a > 0 x+1 f (x) 0, elsewhere. II n 1 Show that | is sufficient for a. X; +1The 4-dimensional random vector X has PDF fX(x)={1 when 0≤xi≤1, i=1,2,3,4 and 0 otherwise}. Are the for components of X independent random variables?
- Let X and Y be independent uniform random variables on (0,1). Let Z = [1/(X+Y)]. (Recall that is the largest integer ≤ x.) (a) Find P(Z = 0). (b) Find E[Z].2. Let U1, U2,... be independent random variables, each with continuous distribution that is uniform in the interval [-3, 3]. Let S, = U ++ Un. (a) Compute the moment generating function of U1. (b) Compute the moment generating function of S,. (c) Determine E(S) and E(S) (by any method).Suppose we have the quadratic function f(x)=A(x^2)+2X+C where the random variables A and C have densities fA(x)=(x/2) for 0≤x≤2, and fC(x)=3(x^2) for 0≤x≤1. Assume A and C are independent. Find the probability that f(x) has real roo
- 2. Assume X ~ Exp(0), so that its pdf is 1 f(x; 8) x > 0,0 < 0 < o. e We have 5 samples, 1,2,3,4, and, 5, under Exp(0). Find the maximum likelihood estimator of 0 using the five samples.-) Suppose X and Y are continuous random variables. The range of X is [1,3], the range of Y is [0, 1]. The joint pdf of X and Y be given by f(x, y) = 2xy³ - 2y³. Verify if X and Y independent random variables.Let the joint pdf of random variables X,Y be fx,y (x, y) = a(x + y)e-2-Y, for all æ > 0, y 2 0. Find the conditional pdf Tylx (y|x) = &x2) fx(x)