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- 34. Consider the continuous random variable X whose pdf is given by f(r) = (a-r³)Io<<1(1). (a) Find the value of a that makes f(z) a pdf. (b) Find E(X). (c) Find V(X). (d) Find P(X ≤).Q. 2 For the exponential random variable X with parameter à = 2, compute the ollowing probabilities. (a) P(1 5). (c) P(X > 5|X > 3). (d) P(X > 2|X > 3).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.
- 1. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).An individual has a vNM utility function over money of u(x) = Vx, where x is final wealth. Assume the individual currently has $16. He is offered a lottery with three possible outcomes; he could gain an extra $9, lose $7, or not lose or gain anything. There is a 15% probability that he will win the extra $9. What probability, p, of losing $7 would make the individual indifferent between to play and to not play the lottery? (Make sure to answer in the form, 0.X, i.e. 0.25) Enter your answer hereQ1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 02. Let X and Y be independent random variables with E[X] = 4, Var[X] = 84, E[Y] = 6, and Var[Y] = 54. Find E[5X + 2Y - 18] - b. Find E[-3Y+X+3] -a. -с. Find E[X2] d. Find ox e. Find E[Y2] f. Find Oy g. Find Var[X+3Y - 6] h. Find Var[3X - 2Y + 35] Find E[(2X - 8)/5] j. Find Var[(2X - 8)/5] i.A random variable, X, has a pdf given by The expected value of Y is O a. 3/5 O b. 5/3 O c. 7/5 O A random variable Y, is related to X by Px(x) d. 7/3 = 1 -3 < x < 1. and 0 otherwise 4 Y = X²8. Let the random variable X have the pdf 2 x2 fx (x) = exp %3D - V2n 2 Find the mean and the variance of X. Hint: Compute E (X) directly and E (X²) by comparing the integral with the integral representing the variance of a random variable that is N(0,1). i DCO 04 < (X - 5)2 < 38.4).3. Suppose X is a discrete random variable with pmf defined as p(x) = log10 ( for %3D x = {1,2,3,...9} Prove that p(x) is a legitimate pmf.Q1:Let X be a random variable with pdf f (x) = cx + d for 0 = a- Find the values of constants c and d. b- Find the cdf of X. c- find P(IX| <). P(Suppose that the waiting time X (in seconds) for the pedestrian signal at a particular street crossing is a random variable with the following pdf. · (1 − x/71)² 0 ≤ x < 71 {0 f(x) = {√ √ otherwise If you use this crossing every day for the next 6 days, what is the probability that you will wait for at least 10 seconds on exactly 2 of those days?Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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