Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show that P(Yn = 1) = 1/2, but Yn, n E N is not a Bernoulli...
Let Xi, X2...-Xn be a iid. sample from Bernoulli(p) and let Yn-Σηι(X-P)/n. Show that Ya converges to a degenerate distribution at 0 as n-o.
Let X1 Xn be a random sample of size n from a Bernoulli population with parameter p. Show that p= X is the UMVUE for p. 5.4.22 Let X1 Xn be a random sample of size n from a Bernoulli population with parameter p. Show that p= X is the UMVUE for p. 5.4.22
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let Zn-(Sn-np)/ V np (1-p). Show that as n oo, Zn converges to the standard normal RV Z~ N(0,1) in distribution. (ii) Conclude that if Yn~Binomial(n, p), then (iii) From i, deduce that have the following approximation x-np which becomes more accurate as n → oo.
Exercise 1. Customers at a coffee shop ask for hot, denote it X 0, or cold, denote it Xn 1, beverages according to a Bernoulli process with parameter p. Let N denote the first time that a customer wants the same kind as their predecessor. (1) Find the PMF of N (2) What is the probability that XN+1-1? (3) Cold drinks take 30 seconds to prepare and hot drinks take 60 seconds. What is the expected time taken to serve...
#s 2, 3, 6 2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
Let p E [0,1] with pメ, and let (Xn)n=o b l e the Markov chain on with initia [0,1] given by distribution δο and transition matrix 11: Z Z ify=x-1 p 0 otherwise. Use the strong law of large numbers to show that each state is transient. Hint: consider another Markov chain with additional structure but with the same distribution and transition matrix Let p E [0,1] with pメ, and let (Xn)n=o b l e the Markov chain on with...
& Let Yn = ao Xn ta, x n- do xn + a, Xn-1 ; n =1,2,... where Xi are iid RVs u ; n = 1, 2, with equal moon o and va siance 2.1 95 { yn in 21% SSS? Is {Yn: n 21 } WSS?
1) Customers at a coffee shop ask for hot, say Xn 0, or cold, say X 1, beverages according to a Bernoulli process with parameter p. Let N denote the first time that a customer wants the same kind as their predecessor. (a) Find the pmf of N. (b) What is the probability that XN+ 1? (c) If cold drinks take 30 seconds to prepare and hot drinks take 60 seconds. What is the expected time taken to serve the...
1. Let X1, ..., Xn, Y1, ..., Yn be mutually independent random variables, and Z = + Li-i XiYi. Suppose for each i E {1,...,n}, X; ~ Bernoulli(p), Y; ~ Binomial(n,p). What is Var[Z]?
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...