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- Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; isSuppose 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 roo3. Let Y be the number of speeding tickets a YSU student got last year. Suppose Y has probabilitymass function (PMF)y 0 1 2 3fY (y) 0.12 0.13 0.33 0.42(a) What is the probability a YSU student got exactly one ticket?(b) What is the probability a YSU student got at least one ticket?(c) Compute µY , the mean of Y .(d) Find the variance and standard deviation of Y .(e) What is the probability that Y exceeds its mean value?
- An ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.Let X be a random variable and a real number. Show that E(X - a)² = varX + (µ − a)² Hereμ = EX is the expected value of the random variable X and varX = E(X - μ)^2 is the variance of the random variable X. Guidance: start from the representation - (X-a)^2 = (X µ + μ- a)^2 and group the right side of the representation appropriately into the form (Z + b)^2, where Z is some random variable and b is a real number and open the square. The task should be solved with the help of the expected value calculation rules.Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable, X for the right tire and Y for the left tire, with joint pdf F(x, y) = { K (x² + y²) 20 ≤ ≤ 30, 20 ≤ y ≤ 30 otherwise (a) What is the value of K? (Enter your answer as a fraction.) K= (b) What is the probability that both tires are underfilled? (Round your answer to four decimal places.) (c) What is the probability that the difference in air pressure between the two tires is at most 2 psi? (Round your answer to four decimal places.) (d) Determine the (marginal) distribution of air pressure in the right tire alone. for 20≤x≤ 30 (e) Are X and Y independent rv's? Oves, f(x, y) = fx(z) - fy (y), so X and Y are independent. Oves, f(x, y) + fx(z) - fy(y), so X and Y are independent. ONO, f(x, y) = fx(2) - fy(y), so X and Y are not independent. ONO, f(z,y) #fx(z) - fy(y), so X and Y are not independent.
- Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 24 psi. Suppose the actual air pressure in each tire is a random variable-X for the right tire and Y for the left tire, with joint pdf will be f(x, y) = K (x² + y²) if 20 ≤ x ≤ 30, 20 ≤ y ≤ 30 Are X and Y independent random variables? In order to answer this question, you need to find the value for K first. X and Y are not independent. X and Y are independent. cannot decide none of the other option is correct2. Let f(x) = 3x(x² − 1) (x³ + 2x + 3) (a) Use distribution to expand the function. (b) Use the rules in from the table above to differentiate the function.2. Let Y,,., Y, be independent random variables such that Y, (Yı., Yp)" and 0 = (0,.,0p)". Let = 0(Y) = (0,(Y),... , @p(Y))" be an estimator of 0, and let g(Y) = (g(Y),... , gp(Y))" = – Y. Denote by || - || the Euclidean norm, ||Y° = Y} + .. + Y. N(8), 1). Write Y = %3D Suppose that D(Y) = @g(Y)/ay, exists. Then it is known that %3D R(Ô(Y)} = +2 D(Y) + É19(Y)² =1 is an unbiased estimator of the risk of 0, under squared error loss L(0, ê) = ||0 – e|P. [You are not required to show this]. %3D (i) The James-Stein estimator is 6.s(Y) = (1 – )Y. _P-2y ||Y? Show that an unbiased estimator of the risk of d Js(Y) is Řlójs(Y)) = p – (p - 2) /||YII°. Deduce that Y is inadmissible as an estimator of 0. Is ô js(Y) admissible? Justify your answer.
- 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, ..., X, be independent random variables and Y = min{X1, X2, ..., Xm}. Fy (y) = 1 – || (1 – Fx,(y)) i=1 (a) A certain electronic device uses 5 batteries, with each battery to have a life that is exponentially distributed with mean of 48 hours and is independent of the life of other batteries. If the device fails as soon as at least one of its batteries fail, what is the expected life of the device?Let X be a random variable with pdf: f(x)= k(10 - 2x) for x being [0,5] a.) Find k b.) Find E(X) c.) Find Var(X) d.) Find E(3 *squareroot(X))