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- LetXbea randomvariable havingthe uniformdistribution ontheinterval(θ,θ+1),θ∈R.Showthat Xis not complete.Find the moment generating function MU(t) for the standard uniform random variable U (the continuous random variable whose density function is 1 on [0,1] and 0 elsewhere)The probability that Paul’s train to work is late on any day is 0.15, independently of other days. (i) The number of days on which Paul’s on which Paul’s train to work is late during a 450-days period is denoted by the random carriable Y. Find a value of a such the P(Y>a)≈ 1/6 . In the expansion of (0.15+0.85)^50 , the terms involving 0.15^r and 0.15^(r+1) and denoted by Tr and Tr+1 respectively. (ii) Show that Tr / Tr+1 = 17(r+1)/3(50-r)
- The area under the curve between z=-1.32 and z=2.12Consider a random process X(t) defined by 4 X(t) = Ucost + (V + 1)sint, where U and V are independent random variables for which E(U) = E(V) = 0; E (U²) = E (V²) = 1 a) Find the auto covariance function of X(t) b) Is X(t) wide sense stationary? Explain your answer Jf the Wwss process (Y(+ is given by Y(+) - 100 cos(10t where uniformlsFour refills for a ballpoint pen are selected at random from a box that contains 3 blue refills, 2 red refills and 3 green refills. If X is the number of blue refills and Y is the number of red refills selected, finda. The joint probability mass function p(x, y), andb. P[(X, Y) ∈ A], where A is the region {(x, y)|x + y≤1}
- Evaulate the integral with a triangle, T and vertices (0,1) (1,0) and (1,1)s In a certain town, the daily consumption of natural gas (in millions of m3) follows approximately agamma distribution with α = 2 and β = 4. If the daily capacity of that town is 8 million m3 of natural gas, whatis the probability that on any given day the natural gas supply is inadequate?(Hint: Use integration by parts forthe integral)□ 0.51 □ 0.61 □ 0.71 □ 0.81 □ 0.91Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50
- Let f(x) = 1/[π(1 + x2)], −∞< x < ∞ be the pdf of the Cauchy random variable X. Showthat E (X) does not exist. (hint: split into two integrals (−∞,0) and (0,∞))Let Xi be IID random variables which have the same law as X. Let L(t) = E(e^tX.) Suppose that this is well defined for t ∈ [−1, 1]. Express the moment generating function of the Sum from i=1 to k Xi in terms of k and LFind the density of Z = X-Y for independent Exp(λ) random varibles X and Y. Show all Work!!