10) a) Suppose A= 0,1,2,3,4 and P = {0,3}, {1,2},{4) is a partition of A. Identify the relation Rinduced by P on A. b) Suppose A = 0,1,2,3,4) and R= (0,0),(0,4),(1,1),(1,3),(2,2),(3,1),(3,3),(4,0),(4,4) is an equivalence. relation on A. Find the partition P induced by R.
25. Suppose that n > 2. Prove that Sn is generated by each of the following sets. (a) Ti={(1,2), (1 ,2,3,... ,n)) (b) T2={(1,2,3 ,... ,n-1), (n - 1,n)} (c) Ts={(1,2), ( 2,3), (3,4),... , (n - 1, n)} (d) Ta={(1,n), (2, n), (3, n),... , (n - 1, n)} 72-
Problem 5.4 (10 points) Let (Sn)n-01. be a simple, symmetric random walk with starting value So-s e R. (a) Show that ES for alln0 b) Show that ElSn+1 Sn] Sn for 0. (c)Suppose that (Sn)n-0,12,. . denotes the profit and loss from $1 bets of a gambler with initial capital So-s who is repeatedly playing a fair game with 50% chances to win or lose her stake. What are the interpretations of the results in (a) and (b)?
Problem 5.4...
(5) Recall that X ~Uniform(10, 1,2,... ,n - 1)) if if k E (0, 1,2,... ,n -1, P(x k)0 otherwise (a) Determine the MGF of such a random variable. (b) Let X1, X2, X3 be independent random variables with X1 Uniform(10,1)) X2 ~Uniform(f0, 1,2]) Xs~ Uniform(10, 1,2,3,4]). X3 ~ U x2 ~ Uniform(10, 1,2)) 13Uniform Find the laws of both Y1 X1 +2X2 +6X3 and Y2 15X1 +5X2 + X3. (c) What is the correlation coefficient of Yi and ½?...
The moment generating function (MGF) for a certain probability distribution is given by 2 (2 + 2) , M(t) = R. t 2 Suppose Xi, X2, are iid random variables with this distribution. Let Sn -Xi+ (a) Show that Var(X) =3/2, i = 1,2. (b) Give the MGF of Sn/v3n/2. (c) Evaluate the limit of the MGF in (b) for n → 0.
The moment generating function (MGF) for a certain probability distribution is given by 2 (2 + 2)...
Suppose (Sn) is a sequence such that, for all n 1, we have 1 Sn+1 – snl < n2 Show that (sn) converges
4. Given the pd )if -1,2 0 otherwise. (a) Determine the proper value for c. (b) Find P(0 < X<1) (c) Find E(X) (d) Find E(X) and Var(X)
4. Given the pd )if -1,2 0 otherwise. (a) Determine the proper value for c. (b) Find P(0
x Incorrect. The following function is probability mass function. f(x) = 2x+4 40 x=0,1,2,3,4 Determine the mean, μ, and variance, σ2, of the random variable. Round your answers to two decimal places (e.g. 98.76). 2.7
Suppose the andom variable X has pmf 3 , n 0, 1,2 Find the median and the 70th percentile.
Find the imin and limsup oP the followi uences denoted lou Sn n+1 (b) sn = n (1+(-4)^) + n.)((-1) ) (c) sn=봬 , where ynis - bounded (d) sn =n2. sequence.
Find the imin and limsup oP the followi uences denoted lou Sn n+1 (b) sn = n (1+(-4)^) + n.)((-1) ) (c) sn=봬 , where ynis - bounded (d) sn =n2. sequence.