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- (3) Let X = b >(8,-) find E(5+6x) and distribution function.The moment generating function of the random variable X is given by: M(t) = e^(12t+24.5t^2). a. How is X distributed? b. Write the complete pdf of X. c. Find P(X>28.8). d. Find P(1.5 < X ≤ 22.5). e. Find P(0.784 < (X-12)^2 < 188.209).Suppose the inter-arrival time of people using the elevator can be modeled as X ~ Exp(1/6) (in minutes). What is the distribution of the number of people waiting at an elevator in two minutes (since the last elevator left)? Hint: Let Tn be the time when the n-th person arrives at the elevator, and let N be the number of people waiting at time t, then P(N>=n) if and only P(Tn =n) and P(N>=n+1). The difference of the two gives you P(N=n).
- Let X1 and X2 be independent random variables from a normal distribution with mean equal to one and standard deviation equal to one. Derive the distribution of (X2 + X1-2)^2/(X2 - X1)^2. If the resulting distribution is one of the special distributions, identify the distribution and all relevant parameters; otherwise, state "unknown".Use the geometric distribution to derive E(Y), E(Y^2), and V(Y) from the Poisson distribution3) Thas an exponential distribution such that P(TS2)=2P(T>4) What is Var(T)=? 4)
- Let X1 and X2 be IID exponential with parameter > 0. Determine the distribution ofY = X1=(X1 + X2).Simplify the likelihood ratio e (µ1,..., Hr, o2) max (µ1,...,Hr,o²)EO0 e (41,...,Hr; o2) ' max (41,...,Hr,0²)EO SSTR/(r-1) SSE/(nt-r) and show that A is a decreasing function of F =Suppose X is a random variable whose pdf is given by f(x) =k(4x-2x^2) 01)
- Suppose X has a continuous uniform distribution over the interval [-2, 2]. Determine (a) Mean (b)variance, and (c) standard deviation of XThe number of clients a store has during a week is Poisson distributed with - 25. The amount of money spent by each client follows a random = el,000t+250t2. Assuming the mean A variable with moment generating function o(t) independence between the number of clients, and the amount of money spent by each client. Find the mean and variance of the amount of money received by the store every week.Suppose that a continuous random variable X follows a uniform distribution over {-4, 4]. (a) Find the PDF of X (b) Use integration to find the distribution function of X and sketch it (c) Using the PDF or the CDF, find P(X< 0)