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- Q1. The cumulative distribution function of the random variable X is F(x) = 0, (x+1)², 1- (1-2)², 1, for x 1 (a) Compute-P(0.5 < X <1)¶ (b) What is the probability density function of X.?5 Suppose the distribution function of the continuous random variable { is 1-es*, x20; F(x)=• 0, x3); (2) the density function f(x)1. A continuous random variable X has a probability density function (PDF) p(x) = k( 8-x^2/2) on the interval [0, 4] (a) Find k such that p(x) is a valid PDF. (b) Find P(X ≤ 1). (c) Find the mean, µ, of X. (d) Find the variance, σ 2 , of X.
- b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x;0)=√√2n0 0, -20₁ if -∞0 < x <∞0 otherwise (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. (iii) Check whether or not the estimator obtained in (ii) is unbiased. (iv) Find the Fisher information in this sample of size n about the parameter 0.b) Let X₁, X2, X3.....Xn be a random sample of n from population X distributed with the following probability density function: ze zo, f(x;0)=√2m0 0, (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. if -∞20. The quantity demanded (in thousands of pounds) of a Red Farms Butter each week is a contniuouis 1 1 2 36 random variable with probability density function (PDF): f(x)=x -x² (0≤x≤6). a) Show that it is a PDF on the specified interval. b) What is the probability that between 3-4 thousand pounds of butter will be demanded in a randomly selected week?Suppose that the distribution function of X is given by //check pic // Calculate the probability mass function of X6. Suppose that the random variables X and Y have joint probability density function given by x+y, 0A continuous random variable X is defined by f ( x ) = (3+x)^2/16 ; -3<x<-1 f(x) = = ( 6 - 2x^2)/16 ; -1<x<1 f(x) = ( 3 - x^2)/16 ; -1<x<3 a. Verify that f(x) is density b. find the meanRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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