Let X and Y be two independent and identically distributed random variables with expected value 1 and variance 2.56. First, find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). Now suppose that X and Y are independent and identically distributed N(1,2.56) random variables. What is P(|X + Y − 2| ≥ 1) exactly? Why is the upper bound first obtained so different from the exact probability obtained?
Let X and Y be two independent and identically distributed random variables with expected value 1...
Let Y, Y2, Yz and Y4 be independent, identically distributed random variables from a population with mean u and variance o. Let Y = -(Y, + Y2 + Y3 +Y4) denote the average of these four random variables. i. What are the expected value and variance of 7 in terms of u and o? ii. Now consider a different estimator of u: W = y + y + y +Y4 This an example of weighted average of the Y. Show...
9. Let X and Y be independent and identically distributed random variables with mean u and variance o. Find the following: (a) E[(x + 2)] (b) Var(3x + 4) (c) E[(X-Y)] (d) Cov{(X + Y), (X - Y)}
Let X and Y be two independent and identically distributed random variables that take only positive integer values. Their PMF is pX(n)=pY(n)=2−n for every n∈N , where N is the set of positive integers. Fix a t∈N . Find the probability P(min{X,Y}≤t) . Your answer should be a function of t . unanswered Find the probability P(X=Y) . unanswered Find the probability P(X>Y) . Hint: Use your answer to the previous part, and symmetry. unanswered Fix a positive integer k...
Let X 1 and X 2 be statistically independent and identically distributed uniform random variables on the interval [ 0 , 1 ) F X i ( x ) = { 0 x < 0 x 0 ≤ x < 1 1 x ≥ 1 Let Y = max ( X 1 , X 2 ) and Z = min ( X 1 , X 2 ) . Determine P(Y<=0.25), P(Z<=0.25), P(Y<=0.75), and P(Z<=0.75) Determine
6. (15 pts.) Let X,X.. Xn be independent and identically distributed erponentially distribu random variables, each with mean ux 1. Let a. Calculate E[W] b, Calculate ơw, the variance of W c. Calculate the probability P[X, Ss 1 d. Approximate the probability P[W 1] when n is large e. Suppose n - 1000, and you have one guess at what W is. What numerical value would you pick? What fundamental result in probability theory are you basing your answer on?
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when 6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
(a) Suppose that Xi, X2,... are independent and identically distributed random variables each taking the value 1 with probability p and the value-1 with probability 1-p For n 1,2,..., define Yn -X1 + X2+ ...+Xn. Is {Yn) a Markov chain? If so, write down its state space and transition probability matrix. (b) Let Xı, X2, ues on [0,1,2,...) with probabilities pi-P(X5 Yn - min(X1, X2,.. .,Xn). Is {Yn) a Markov chain and transition probability matrix. be independent and identically distributed...
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(H = 0) = q = 1-p. (Such random variables are called Bernoulli random variables.) a Find the moment-generating function for the Bernoulli random variable Y b Find the moment-generating function for W = Yit Ye+ … + . c What is the distribution of W? 1.
Problem 42.5 Let X and Y be two independent and identically distributed random variables with common density function f(x) 2x 0〈x〈1 0 otherwise Find the probability density function of X Y. 42.5 If 0 < a < l then ÍxHY(a) 2a3. If 1 < a < 2 then ÍxHY(a) -릎a3 + 4a-3. If a 〉 2 then fx+y(a) 0 and 0 otherwise.