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- (1) Let X b(16,- 5,-) find E(4- 3x) and distribution function."Time headway" in traffic flow is 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. Let X = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (sec). Suppose that in a particular traffic environment, the distribution of time headway has the following form. x >1 f(x) =10 xs1 (a) Determine the value of k for which f(x) is a legitimate pdf. (b) Obtain the cumulative distribution function. x> 1 F(x) = xs1 (c) Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to four decimal places.) Use the cdf from (b) to determine the probability that headway is between 2 and 3 sec. (Round your answer to four decimal places.)"Time headway" in traffic flow is 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. Let X = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (sec). Suppose that in a particular traffic environment, the distribution of time headway has the following form. x> 1 f(x) = (a) Determine the value of k for which f(x) is a legitimate pdf. (b) Obtain the cumulative distribution function. x> 1 F(x) = xs1 (c) Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to four decimal places.) Use the cdf from (b) to determine the probability that headway is between 2 and 3 sec. (Round your answer to four decimal places.) (d) Obtain the mean value of headway and the standard deviation of headway. (Round your standard deviation to three decimal places.) mean standard deviation (e) What is the probability that headway is…
- "Time headway" in traffic flow is 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. Let X = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (sec). Suppose that in a particular traffic environment, the distribution of time headway has the following form. f(x) = (a) Determine the value of k for which f(x) is a legitimate pdf. F(x) = k x12 0 (b) Obtain the cumulative distribution function. 0 x > 1 x ≤ 1 mean x > 1 (c) Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to four decimal places.) x ≤ 1 Use the cdf from (b) to determine the probability that headway is between 2 and 3 sec. (Round your answer to four decimal places.) standard deviation (d) Obtain the mean value of headway and the standard deviation of headway. (Round your standard deviation to three decimal places.) (e) What is the probability…(3) Let X = b >(8,-) find E(5+6x) and distribution function."Time headway" in traffic flow is 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. Let X = the time headway for two randomly chosen consecutive cars on a freeway during a period o heavy flow (sec). Suppose that in a particular traffic environment, the distribution of time headway has the following form. x! f(x) = (a) Determine the value of k for which f(x) is a legitimate pdf. (b) Obtain the cumulative distribution function. F(x) = x< 1 (c) Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to three decimal places.) 016 Use the cdf from (b) to determine the probability that headway is between 2 and sec. (Round your answer to three decimal places.) .014 (d) Obtain the mean value of headway and the standard deviation of headway. (Round your answers to three decimal places.) mean .6 standard deviation (e) What is the probability that headway is within 1…
- Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; isThe cumulative distribution function F(x) of a random variable X satısfies F(-1) = 0.25, F(0) = 0.33, F(0.8) = 0.72, F(3.2) Find the value of F(2). |(Hint: use the fact that F(x) is a non-decreasing function. That is, if x < y, we have F(x) < F(y).) = 0.72, F(9.9) = 1.3. Let Y be the number of speeding tickets a YSU student got last year. Suppose Y has probabilitymass function (PMF)y 0 1 2 3fY (y) 0.12 0.13 0.33 0.42(a) What is the probability a YSU student got exactly one ticket?(b) What is the probability a YSU student got at least one ticket?(c) Compute µY , the mean of Y .(d) Find the variance and standard deviation of Y .(e) What is the probability that Y exceeds its mean value?