(a) Consider the following poisson mass function for the future lifetime of a newborn: -λ f₁(k) = 2^e^/k! for all k ≥ 0, where k is a discrete random variable (i) Find an expression for S (k). (ii) Find an expression for F (k). (iii) For λ = 3 calculate F (8).
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- Q.1 A continuous random variable X has the following function -3The moment generating function of the exponential distribution with rate A is m(t) = for t < A. Moreover, skewness of a random variable X is defined as X – E(X) E /var(X) Find the skewmess for the standard exponential distribution.Let X and Y be independent exponential random variables with parameter 1. Find the cumulativedistribution function of Z = X/(X + Y )Find the generating function of the negative binomial mass function = (x =1) f(k) = p' (1-p)k-r, k = r,r +1,..., where 0 < p < 1 and r is a positive integer. Deduce the mean and variance.1. A continuous random variable X has a probability density function (PDF) p(x) = k( 8-x^2/2) on the interval [0, 4] (a) Find k such that p(x) is a valid PDF. (b) Find P(X ≤ 1). (c) Find the mean, µ, of X. (d) Find the variance, σ 2 , of X.two random independant variables X and Y with distributions: X ∼ Poisson(λ) og Y ∼ Poisson(2λ), andobservations x = 2 og y = 5 .(a) What is the log-likelihood function? b) calculate MLE for the observated samplesFind the value of k for the probability mass function 4 f(x) = k () , ) for x = 0,1,2. %3| 4 -(b) A random variable X has a probability density function ГАx (6 — х)* ,0 sxs6 f(x) = |0, otherwise (i) Find the value of the constant A (ii) Calculate E(X), mode and the var(X)3) Assume that a random variable X has moment generating function as follows: e' + e²¹ M(t)=4-2e¹ ∞ Calculate the mean and variance of X. for tRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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