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- QUESTION 6 Suppose we know that X and Y are random variables with var(X) = 2, var(Y) = 3, and cov(X,Y) = -1. Find var (2X – Y). Round to 4 decimal places if needed.A null and alternative hypothesis are given. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. Ho: Ha: p≥ 0.3 р < 0.3 What type of test is being conducted in this problem? A. Right-tailed test B. Left-tailed test C. Two-tailed test4. Question 4: Y is a discrete random variable with PMF S1/3, y=1 p(y) = 2/3, y = 2 (a) Find the MGF for Y. (b) Develop the first two raw moments from MGF and find the variance V(Y). Answers:
- 1. The time to failure of an AC compressor is a random variable T, with the following pdf. 200 f(t) = (10 + t)3 (a) Find R(t). (b) Determine the reliability for the first year of operation. (c) Find the probability of failure occurring in the interval of time [15,20]. (d) Find the design life if a reliability of 0.99 is desired.What would be the correct way of doing this?Question 1 1. Suppose you are given that a continuous random variable has a moment generating function of 1 M (t) = 1-3.2t Find the mean and variance of the random variable. mean = var =
- Question 5 Suppose X is a normal random variable with mean and variance o². (i) If a and b are constants, show that aX + b has a normal distribution and find its mean and variance. (ii) If Z = (X − μ)/o, deduce that Z~ N(0, 1). (iii) Using (ii) find P(X 0 is a constant, show that P(µ – co < X < μ+co) does not depend on μ or o.Pls do it fast and correct.9.
- QUESTION 3 Suppose that the continuous random variables X and Y have the joint density function 4xy x, y = [0, 1] 0 otherwise f(x, y) = and suppose the expected values of Y is E(Y) Round to 4 decimal places if needed. = 3. Find the variance of Y.1-)The probability intensity function and function are integrated thye random variable with normal distrubition W. normal distribution probability density. function integral equals 1. ww MWW w N wwww ww w w w wm +00 1 | fw (w) dw = e-(w-H)²/20² dw = 1 %3D %3D -08 - 00 a)x = ( w -u)/a the accuracy of the statement below, using the equality show. www 1 e dx V2n -00 b) Verify that the following statement is true. I +00 12 = -| e-(x*+y°)/2 dxdy c) show that the integral equals 1 by typing the integral in polar coordinates 12 MAW MA wwMQuestion 12. Let X, Y be two random variables and take values in {0,1} and whose joint distribution is given by: Joint Distribution Y = 0 Y=1 fx(x) X = 0 0.25 0.25 0.5 X = 1 0.25 0.25 0.5 fy(y) 0.5 0.5 1 Mark the choice that is FALSE: (a) E(X² + Y2) = E(X) + E(Y) (b) X and Y are statistically independent. (c) E(X²) = (E(X))² (d) var(X) = var(Y) (e) P(X = 1|Y = 1) = P(X = 1|Y = 0)