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- 6. (Sec. 5.1) Two headlights of a car have the following joint pdf for their useful lifetimes X (the left headlight) and Y (the right headlight) ze(y+1) for r> 0.y > 0 f(x, y) 0 otherwise (a) What is the probability that the lifetime X of the left headlight exceeds 2.8? (b) Find the marginal pdfs of X and Y. Are the two lifetimes independent? Justify your answer (c) What is the probability that the lifetime of at least one headlight does not exceed 2.8?3. (a) Let X be a random variable with PDF f(x) = 1 |x|, |x| ≤ 1. Obtain the MGF of X. (b) Find the PDF of a random variable Y with MGF for t0, and My (0) = 1. 2 et My(0) = (^=¹)'.Assume that Y has a beta distribution with parameters a and ß, so that the pdf of Y is I(a + B) T(a)(3) ³ -ya-¹(1−y)³-¹, 0Suppose that Xi ∼ Gamma(αi , β) independently for i = 1, . . . , N. The mgf(moment generating function) of Xiis MXi(t) = (1 − (t/β) )−αi . (a)Use the mgf of Xi to derive the mgf of ∑i=1 Xi . Determine the distribution of ∑i=1 Xi based on its mgf.68. Let X be a continuous random variable with pdf (a) Find the pdf for Y = √X. (b) Find the pdf for U = ln X. fx(x) = 3x²I(0,1)(x)Let X and Y have the joint pdf f(x,y)= x+y , 0<=x<=1, 0<=y<=1. Find the marginal fx(x) and fy(y). Show if f(x,y) dependent. Compute mean(x), mean(y), variance (x) and variance(y)70. Let X~ Xn. (a) Find the moment generating function mx (t) for X. (b) Use the mgf for X to derive the formula for the mean of X, μ = (c) Use the mgf for X to derive the formula for the variance of X, o² = V(X) = E(X²) - [E(X)]² = m (0) - [m'x (0)]². E(X) = m'x (0).3. If X and Y are jointly continuous random variables with joint PDF fx.y (r, y) = exp(-y)I0.v)(x)I(0,0) (y) (a) Find the joint MGF of X and Y (b) Find the marginal MGFS of X and Y (c) Identify the marginal distribution of X and Y (d) Find the E(XY) using the joint MGF of X and YA CI is desired for the true average stray-load loss μ (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with = 2.1. (Round your answers to two decimal places.) (a) Compute a 95% CI for μ when n = 25 and x = 56.0. watts (b) Compute a 95% CI for μ when n = 100 and x = 56.0. watts (c) Compute a 99% CI for μ when n = 100 and x = 56.0. watts (d) Compute an 82% CI for μ when n = 100 and x = 56.0. watts (e) How large must n be if the width of the 99% interval for u is to be 1.0? (Round your answer up to the nearest whole number.) n =SEE MORE QUESTIONS