A discrete-time random process is defined by X, =s", forn ≥0, where sis selected at random from the interval (0,1). Is X₁ an independent stationary increment (ISI) random process? Justify your answer.
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- = Customers arrive at a certain facility according to a Poisson process of rate > 5 customers/hour. Suppose that it is known that 10 customers arrived in the first 3 hours. Determine the mean total waiting time E[W₁ + W₂+ · + W₁0 | N(3) = 10] : = 55/15 hoursTwo random variables X and Y have joint char- acteristic function l@,,m)=exp(-20,?-80;) (i) Show that X and Y are zero mean ran- dom variables.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).
- 5. Suppose random observations Y₁ and Y2 are independent each distributed N(0, 1). Also let the following random Y² + Y22 variable be defined as s² = Show that 2s2~ 2If X is a continuous random variable with distribution, f(x) =+k if0sxs3 0 elsewhere Determine P(1 SIS2)(11) Let X be a random variable with p.d.f. 4k 1< x<10 f (x)=- , find k. O.w
- You have 1000 dollars to put in an account with interest rate R, compounded annually. That is, if X, is the value of the account at year n, then X, = 1000(1 + R)", for n = 0,1, 2, -... The value of Ris a random variable that is determined when you put the money in the bank, but it does not not change after that. In particular, assume that R~ Uniform(0.04, 0.05). a. Find all possible sample functions for the random process {Xn,n = 0, 1,2, ...}. b. Find the expected value of your account at year three. That is, find E[X3].A random variable Y has the Gaussian distribution on R with mean u and variance o?, Select all the necessarily true statements from the following. Select one or more: O a P(Y > y) y) y) < erp(-(y-4)²/(2a2)) for all y E R. 2no O d. P(Y 2 y) sexp(-(y-p)/(20)) for all y 4 O e. P(Y 2 y) < erp(-(y-)/(20²)) for all y 2 u and P(YIf X1, X2, ...Xn is a random sample from N(0, σ^2), How would you show the following?Suppose the random variable XX is uniform on the interval from 20 to 70, so X∼U(20,70).X∼U(20,70). Compute answers to two decimal places. μxμx == σxσx == P(x=46)=P(x=46)= P(32<x<45)=P(32<x<45)= P(x>45∣x>32)=P(x>45 ∣ x>32)= The 7070th percentile is8) A random variable X is uniformly distributed on the interval (-5, 15). Another random variable Y = e-X/5 is formed. The value of E[Y] is: а. 2 b. 0.667 с. 1.387 d. 2.96754. A manufacturer finds the average demand per day for the machines to repair his new product is 1.5, over a period of one year and the demand is distributed as Poisson variate. He employes two machines. On how many days in a year i) both the machines will be free ii) some demand is refused.