Poisson distribution By using the following probability distribution w(x)e-* E[w(x)]x! p(x,2) = And suppose w (x) =( a – A*) Prove 1-moment generating function equal e a Find 2-factorial moment
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- 1 Expectation of a function of random variable: Find E(πcosπx) for X uniformly distributed on [0,1].Lost-time accidents occur in a company at a mean rate of 0.3 per day. What is the probability that the number of lost-time accidents occurring over a period of 7 days will be exactly 3? Assume Poisson situation. P(X=3)The Poisson distribution gives the probability for the number of occurrences for a "rare" event. Now, let x be a random variable that represents the waiting time between rare events. Using some mathematics, it can be shown that x has an exponential distribution. Let x > 0 be a random variable and let β > 0 be a constant. Then y = 1 β e−x/β is a curve representing the exponential distribution. Areas under this curve give us exponential probabilities. If a and b are any numbers such that 0 < a < b, then using some extra mathematics, it can be shown that the area under the curve above the interval [a, b] is the following. P(a < x < b) = e−a/β − e−b/β Notice that by definition, x cannot be negative, so, P(x < 0) = 0. The random variable x is called an exponential random variable. Using some more mathematics, it can be shown that the mean and standard deviation of x are the following. μ = β and σ = β Note: The number e = 2.71828 is used throughout…