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- I1. The probability distribution function, P(X-x) of a discrete random variable X is defined as for =0 P(X -x) 4. for x=1 for =2 a) Find the cumulative distribution of X, Fx) b). Find i) P(X-1) ii) P(04. For a Poisson distribution X~P(A) (A > 0), which one of the following values does equal to? A. 1 B. 1+ C. 1- D. E(X²) [E(X)² ()2. Let X have a Poisson distribution with mean of 4. Find a) P(23) c) P(X<3)(2). Let st have the distribution of X ~N(4,16). (a). P(x >xo) = 0.05, (b). Fmd the number Xo smce P(X>-x) - 0.952.) Use the moment generating function you just proved to verify that for a Poisson distribution, E(X) = λ and Var(X) = λ.A distribution is given as X ~ U (2, 10). If you draw a graph of the situation, what is the height of the graph? (Express your answer as a decimal.) What is P(3 < x < 6)? (Express your answer as a decimal.) What is P(x = 4)? (Express your answer as a decimal.) What is P(x < 7)? (Express your answer as a decimal.) What is the 75th percentile? What is the 40th percentile? What is the mean? What is the standard deviation? (Round to 2 decimal places.)3. If X has a Poisson distribution and if P(X = 0) 30.25, find P(X > 4)."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 0 x ≤ 1 (a) Determine the value of k for which f(x) is a legitimate pdf. 5 f(x) (b) Obtain the cumulative distribution function. F(x) -5 1-x 0 X> 1 x ≤ 1 (c) Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to four decimal places.) 0.0156 x Use the cdf from (b) to determine the probability that headway is between 2 and 3 sec. (Round your answer to four decimal places.) 0.0271 (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…