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- (2.d) We are interested in the function of X defined by Y = g(X) = X². What is CDF of Y in terms of CDF of X? (Note, in %3D the answer box SQRT=Square Root). Referring to Y = g(X) = XWhat is the expected value (2.e) of the function of the random variable, E[Y?Q1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 0b) A continuous random variable X has the p.d.f f(x) = {A(2 – x)(2 + x), 0 < x < 2, Find (i) the value of A, (ii)P(X <1)(iii) P(1 < X <2). l0, otherwis ------------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 = X²A random sample of size n₁ = 14 is selected from a normal population with a mean of 76 and a standard deviation of 7. A second random sample of size n₂ = 9 is taken from another normal population with mean 71 and standard deviation 11. Let X₁ and X₂ be the two sample means. Find: (a) The probability that X₁ – X₂ exceeds 4. 1 2 (b) The probability that 4.3 ≤ X₁ – X2 ≤ 5.6. Round your answers to two decimal places (e.g. 98.76). (a) i (b) iAnswer no. 1 onlyQ6 Let X be a random variable with mean μ and suppose the expected value E[(X – μ)²k] exists for some k ≥ 1. Show for any d > 0, that P(|X-μ ≥ d) ≤ E[(X − µ)²] d2k (HINT: Use an argument similar to that used to prove Chebyshev's inequality.)A relatively rare disease D occurs with P(D)=0.01. There exists a diagnostic test such that: P(positive test | D)= 0.99 P(positive test | not D)=0.1 Using the bayes rule, what is P(D | positive test)?Suppose the random variable T is the length of life of an object (possibly the lifetime of an electrical component or of a subject given a particular treatment). The hazard function hr(t) associated with the random variable T is defined by hr(t) = lims-o- P(t ≤ TIf F(x) is cumulative distrbution function of a continuous random variable X, the probability (function f(x) is the dervative of F(x (Fa)se (TrụeEach 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) = JK(r² + y²) 0 20≤x≤ 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? OYes, f(x,y) = fx(z) - fy(y), so X and Y are independent. OYes, f(x, y) + fx(x) · fy(y), so X and Y are independent. ONO, f(x, y) = fx(z) fy(y), so X and Y are not independent. ONO, f(x, y) + fx(2) fy(y), so X and Y are not independent.My Find the standard deviation of the discret probability dist x P(x) xp(x)) (x-M)² | (x-mipul 2 σ = √(x-M)² pxx) ટે О 0.21 10*0.21=0 /10-112 =. M= Exp(x) 1 0.30 1*0.3 = 0.3 2 0=49 EXP(X) =MSEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON