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- 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 LConsider a random variable X with E[X] = 10, and X being positive. Estimate E[ln√X] using Jensen’s inequality.Let i_t denote the effective annual return achieved on an equity fund achieved between time (t -1) and time t. Annual log-returns on the fund, denoted by In(1 + i_t) , are assumed to form a series of independent and identically distributed Normal random variables with parameters u = 6% and o = 14%.An investor has a liability of £10,000 payable at time 15. Calculate the amount of money that should be invested now so that the probability that the investor will be unable to meet the liability as it falls due is only 5%. Using only formulas, no tables
- If y is random variable has a probability mass function defined as follows find 1. and 2.Find the moment-generating function of the contin-uous random variable X whose probability density is given by f(x) =1 for 0 < x < 10 elsewhere and use it to find μ1,μ2, and σ2.If Y is a continuous, uniformly distributed random variable over the interval(4,10), then the value of the PDF between 4 and 10 is?
- If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)Let X be a Gaussian random variable (0,1). Let M = ln(5*X) be a derived random variable. What is E[M]?In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Poisson distribution with probability mass function f(x), where f(x) = θ x e −θ/x! and θ is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was.
- 9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y): - 30 -50x²-50y² +80xy for -5. Suppose Z is a Gaussian random variable with mean 0 and 1. (a) Find the moment generating function of Z2. (b) Find the density function of Z2.Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50