25. Use Renyi's theorem to prove that if {Xn, n ≥ 1} is iid with common con- tinuous distribution n P{[Xn = \/ X;] i.o.} = 1. i=1
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![25. Use Renyi's theorem to prove that if {Xn, n ≥ 1} is iid with common con-
tinuous distribution
n
P{[Xn = √ X¡] i.o.} = 1.
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- Let X1 and X2 be IID exponential with parameter > 0. Determine the distribution ofY = X1=(X1 + X2).5. let X1 and X2 have the joint pdf f(x1,x2) = x1+ x2, 0 < x1 < 1, 0 < x2 < 1. Find the conditional mean and variance of X2 given X1 = x1, 0 < x1 < 1.2) The time between successive customers coming to the market is assumed to have Exponential distribution with parameter l. a) If X1, X2, . . . , Xn are the times, in minutes, between successive customers selected randomly, estimate the parameter of the distribution. b) b) The randomly selected 12 times between successive customers are found as 1.8, 1.2, 0.8, 1.4, 1.2, 0.9, 0.6, 1.2, 1.2, 0.8, 1.5, and 0.6 mins. Estimate the mean time between successive customers, and write down the distribution function. c) In order to estimate the distribution parameter with 0.3 error and 4% risk, find the minimum sample size.
- Let X, and X2 be independent exponential distributions with the same parameter 2. What is the joint distribution of Y, = X, + X2 and Y, X1 -? What are the distributions of Y, and X1+X2 Y2?Q3. Generate three random variates according to each of the following distributions using your own uniform random variates from U(0, 1) and list them: • Discrete Uniform: U (15, 17) • Uniform: U (0.57, 1.5). • Triangular: Triang (2, 10, 4). • Binomial: Bin (12, 0.4). • Lognormal: LN (5, 1). Beta: Beta (5, 5). • Negbin (8, 0.4) • Weibull (.5, 2.5) Geom (0.5) • PT5 (0.5, 0.5)4. Let X, Y and Z have the joint pdf x2 + y? + z2 + y?. (- + (2m) -3/2 exp 1+ xyz exp 2 where -o < x < ∞, -∞ < y < ∞ and -o < z < ∞. Show that X, Y and Z are pairwise independent and that each pair has a bivariate normal distribution.
- 5. Find the moment generating function of the exponentially distributed random vari- able with pdf Xe-, x≥0 and use it to evaluate the mean and variance of X.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).(47) Let X z b(8,-) find E(5+6x) and distribution function.
- Let X1, . . . , Xn be iid from Exp(β). Find the distribution of the sample maximum Y = X(n). Is this an exponential distribution? (This question was rejected this morning. I did double check and there is no missing info with this question.)2.5. Let N be a random variable with distribution function 0 if <0 F(2) = 〈a,2 if 0 < x < 1 Find P{1,4 < X < 5), P12 < X < .8}, and P(X = 1/2).7. Let X be a continuous random variable with pdf f(x)=√ 0SEE MORE QUESTIONS