Exercise 3.3 Suppose that the probability density function of X is f (x) = 3x, 0, 0 2/3).
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Q: Exercise 3.4 In Exercise 3.3, the probability density function of X is f (x) = ] 3x?, 0, 0 <x<1…
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A: Explanation of the answer is as follows
Q: Exercise 3.3 Suppose that the probability density function of X is f (x) = ] 3x?, 0, 0<x<1 elsewhere…
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Q: 0 <x < 1 2x, f(x) =D { %3D 0, otherwise.
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- Let x be a continuous random variable that can take any value in the interval [0,3]. Let its probability density function be f(x) = {cx² 0Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3,..., X has the following probability density function (exe-e a) b) c) f(x; 0) = x! 0, " for x = 0,1,2,... elsewhere Show that for any &> 0 and S₂ = 1X₁, lim P(|S₂ - 0 ≥ ) = 0. 12-00 Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 0. Let 0₁ = X₁+²x₂+2x₁-X and Ô₂ = (X₁ + X₂ + X3 + X4) be two unbiased estimators of 8. Which 4 one of the two estimators is more efficient? d) What is the Cramer-Rao lower bound for the variance of the unbiased estimator of the parameter 9? e) Use the one-parameter regular exponential family definition to find the functions, h(x), c(0), w(0) and t(x).The joint density function of two continuous random variables X and Y is fxy(x, y) = (2x+y) if 2Suppose a random variable X has a probability density function given by O elsewhere f (x) = kx (1 − x) 0 ≤ x ≤ 1 0 elsewhere a. Find the value of k that makes this a probability density function b. Find P (0.4 ≤ x ≤ 1) c. Find P(x ≤ 0.4) d. Find P (0.8 ≤ x))The probability density function (pdf) of continuous random variable X is as follows: x<0 0A continuous random variable, X, has the following probability density function. k cos for 0sxs1 f(x)=- for all other values of x (a) Show that k = Find p( x sxs). (b) (c) Find the median of X.Let X will be continuous random variable with probability density function as below. f(x) ={0, otherwise S4x3 (0The time, X, for clearing side effects of water purification, in days, has the following cumulative distribution F: 1 F(x) = 0 for x 9 2 a) What is the probability density function for X for 1 5? c) What is the probability X 7? e) What is the probability that X > 6.3? f) What is the probability that X > 7 given the X > 6.3? g) Calculate the 70th percentile of X. h) What is the expected value of X? i) What is the expected value of x2 ? j) What is the variance of X? k) What is the probability that X is more than 0.1 below its expected value?Recommended 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