show that negative binomial distribution with the following pdf belongs to the exponential family
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show that negative binomial distribution with the following
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- For a certain insured, the distribution of aggregate claims is binomial with parameters m = 12 and q = 0.25. The insurer will pay a dividend, D, equal to the excess of 80% of the premium over claims (excess = 0.8 x premium - claims), if positive. The premium is 5. Calculate E[D].T has an exponential distribution such that P(T≤2)=2P(T>4) What is Var(T)=?Let X ~ U[0,1] and Y = -βln(1-X). What is the distribution of Y? Justify.
- For the continuous probability function f(x ) = kx^2e^-x when 0≤x≤1. Find (a)k (b)mean (c)varianceIf X1, X2, ... , Xk have the multinomial distribution of Definition 8, show that the mean of the marginal distribu-tion of Xi is nθi for i = 1, 2, ... , k.The time for the first widget to be manufactured each morning is random between 1 and 3 seconds with a pdf of f(x)=1/162(x-3)^2(x+6) for 0 < x < 6What is the cumulative distribution function, F(x)? What is the probability of a widget being made earlier than 4 s?Answer to four decimal places What is the probability of a widget being made later than 1.3 s?Answer to four decimal places
- 3. If X has an exponential distribution with λ=2. Calculate the following The value of x such that P(X<x)=0.05 corresponds to x=?If X has the exponential distribution given by: f(x) = 4*(1 - x/2); 1 ≤ x ≤ 2. Find the probability that 1.2 < x < 1.6.suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)
- For binomial distribution, E(X) is the expectation. How to induce E(X^2) in terms of E(X). ThanksIf X has the exponential distribution given by f(x) =0.5 e−0.5x, x > 0, find the probability that x > 1.Q1- What is the probability that the number of heads and are equal it a lait coin is tossed, a)15 times b)20 times Q2- Suppose the joint pdf of X and Y is given by: F xy (x y)={ ex +3y, if 0<x< 1 and 0<y<1 0 otherwise find E(x) . Q2- Suppose X is a random variable with E(X) = 5 and Var(X) = 2. What is E(X)?