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- Show that each function defined as follows is a probability density function on the given interval; then the indicated probabilities. f(x)=12(1+x)3/2;[0,) a. P(0X2) b. P(1X3) c. P(X5)3. Show that the following are probability density functions: 1 (a) f(x) = -2*, x = 1, 2, ..., N, and zero elsewhere 2N+1-2 (b) f(x)=p(1-p)*, x=0, 1, 2, ..., and zero elsewhere; 06. Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-0, 0) is a function f such that f(r) > 0 and (2) = 1. (a) Determine which of the following functions are probability density functions on the (-0, 00). (x-1 00 (b) We can also use probability density functions to find the expected value of the outcomes of the event – if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. rf(x) dr yields the expected value for a density f(x) with domain on the real numbers.) Find the expected value for one of the valid probability densities above.In 1938, a physicist named Frank Benford discovered that the number 1 appears in the first digit of random data more often than the number 2, the number 2 more often than the number 3 and so on. In general, the probability of occurrence of the first digit of a number can be written in the form of a probability function x + 1 P(X = x) = log. X a. Prove it P(X = x) = log ) untuk x = 1,2,3,4...,9 x+1 X x = 1,2,3,4..., 9 is a probability mass function 2 b. Find the cumulative distribution function of X!B- Show that if the following function be probability function: for 04. Show the following function is a legitimate probability mass function. Px(x) = Pr(X = x) = ())*(** " x = 0, 1, 2,..., n.6. Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-oo, 00) is a function f such that f (x) >0 and | f (x) = 1. (a) Determine which of the following functions are probability density functions on the (-0, 00). x-1 00 (b) We can also use probability density functions to find the expected value of the outcomes of the event if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. xf(x) dr yields the expected value for a density f(x) with domain on the real numbers.) Find the expected value for one of the valid probability densities above.Problem 4. Let fx (x) be the probability density function of X, which is given by fx(x) = - -2x ce 0, " x > 2 otherwise (a) Find the value of c to make ƒx a valid probability density function. (b) Calculate the cumulative distribution function (c.d.f.) of X. (c) Calculate P(12 < X ≤ 25) using the c.d.f. from part (b). You do not need to simplify your answer.9- The important properties of the probability density function p(x) was: a- P(x) dx = 0.5 b- p(x) dx = 0 c- L p(x) dx = 1Q6. (a) The continuous random variable X has a probability density function given by 2 1 14. A player rolls a die and stops at the fourth roll or when a six appears, whichever occurs first. Let X be the number of rolls. (a) Find its probability mass function. (b) Calculate E[X] (c) Calculate Var(X).4. "Time headway" In highway traffic flow the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point is called time headway. Now, let X = the time headway for two randomly chosen consecutive cars on a highway during a period of heavy traffic flow. The following probability density function of X is suggested by traffic experts: f(x) = 0.15*e-0.15(x-0.5) for (x ≥ 0.5 sec) 0 otherwise a. Draw f(x) from x = 0 to x= 10 sec. b. Find P(X ≤ 5 sec) and show it on the figure you have drawn. c. What is E[X] and Var[X] ? d. Find Cumulative Distribution Function of X.SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,
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