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- 2. where The random variable, X, has a probability density function (pdf), f(x), and zero otherwise. (a.) (b) x+1, f(x) = {1x, 1 81 -1/2 < x < 0 0b) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.Consider the following probability density function of a random variable X f(x) c(x+1)², if −1≤ x ≤ 1, {ox 0, otherwise. (a) What is the value of c such that f (x) is a valid probability density function? (b) What is the mean and variance of X? (c) Let Y = X2, what is the mean of Y? =The total number of hours, measured in units of 100 hours, that a family uses a vacuum cleaner in a period of one year is a continuous random variable X that has the following density function:X que tiene la siguiente función de densidad: 0Suppose the joint distribution of the two random variables X1 and X2is given by S6x2, 0< x2 < x1 < 1, (o, f(x1, 22) = - elsewhere. (a) Find the marginal distribution fx, (x1) of X1 and verify that it is a valid density function. (b) What is the probability that X, is less than 0.5, given that X, is 0.7?Suppose that the amount of time a hospital patient must wait for a nurse's help is described by a continuous random variable with density function f(t) = e-t/3 where t≥ 0 is measured in minutes. (a) What is the probability that a patient must wait for more than 4 minutes? (b) A patient spends a week in the hospital and requests nurse assistance once each day. What is the probability that the nurse will take longer than 5 minutes to respond on (exactly) two occasions? (c) What is the probability that on at least one call out of seven, the nurse will take longer than 7 minutes to respond?The proportion of people who respond to a certain mal-order solicitation is a random variable X having the following density function. 2(x+ 3) 0Consider the probability density fx (x) = a. eb x! where X is random variable whose allowable values range from x = -o to + o. Find (a) the CDF (b) the relation between a and b (c) the probability that x lies between 1 and 2.For a laboratory assignment, if the equipment is working, the density function of the observed outcome X is f(x) = 2x(1-x), 0 < x < 1 Find the variance and standard deviation of X3. Let X and Y be two independent random variables with identical probability density function given by e- for x > 0 f(x) = elsewhere. What is the probability density function of W= max{X, Y } ?Suppose that X and Y have a continuous joint distribution for which the joint probability den- sity function is defined as follows: fx.x (@, y) = { 3y² for 0≤x≤ 2 and 0 ≤ y ≤ 1 otherwise. (a) Determine P(X < 1,Y ≥ 1/2). (b) Determine the marginal probability density functions of X and Y. (c) Are X and Y independent? Explain.c) Let Y₁, Y₂,..., Yn be a random sample whose probability density function is given by f(v:B)= 684 - fa 00 0, elsewhere 200 200 200 and suppose that n = 200, y = 20, y = 100, y = 250 and $ = 0.025. i=1 i) Derive the standard error of ß, se(B) = 0.0009, using MLE approach. ii) Find an approximate 95% Confidence interval for B.SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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