Let X be a random variable with range {0, 1, 2, . . .}. The discrete version of the formula from problem 2.14 in the textbook, i.e. 2.14 (b), can be written
Use this formula to solve the following problem.
A fair dice is thrown n times. The sample space is S = {1, 2, . . . , 6} n , the outcomes are of the form s = (s1, . . . , sn) ∈ S. Let Yn denote the largest of the results thrown, i.e. Yn is a r.v. with Yn(s1, . . . , sn) = max1≤k≤n sk. Find EYn and show limn→∞ E(Yn) = 6.
Homework Answers
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![P(, 29 ) = K ( S < 9 ) s, y = s, e(, , , 211] - f (s, <) - - 6 1. Р (Ү7 )) - - - пEC , ) = P (Y, , K) ИС](http://img.homeworklib.com/questions/673f8110-ae77-11ea-b871-3d3c5a205269.png?x-oss-process=image/resize,w_560)
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Let X be a random variable with range {0, 1, 2, . . .}. The
discrete...
Problem 4 Let X be a discrete random variable with probability mass function fx(x), and let...
Problem 4 Let X be a discrete random variable with probability mass function fx(x), and let t be a function. Define Y = t(X): that is, Y is the randon variable obtained by applying the function t to the value of X Transforming a random variable in this way is frequently done in statistics. In what follows, let R(X) denote the possible values of X and let R(Y) denote the possible values of To compute E[Y], we could irst find...
Let X be a discrete random variable, and let Y X (a) Assume that the PMF...
Let X be a discrete random variable, and let Y X (a) Assume that the PMF of X is Ka2 0 if x- -3, -2,-1,0,1,2,3 otherwise, where K is a suitable constant. Determine the value of K. (b) For the PMF of X given in part (a) calculate the PMF of Y (c) Give a general formula for the PMF of Y in terms of the PMF of X
Let X be a discrete random variable with values in N = {1, 2,...}. Prove that...
Let X be a discrete random variable with values in N = {1, 2,...}. Prove that X is geometric with parameter p = P(X = 1) if and only if the memoryless property P(X = n + m | X > n) = P(X = m) holds. To show that the memoryless property implies that X is geometric, you need to prove that the p.m.f. of X has to be P(X = k) = p(1 - p)^(k-1). For this, use...
Let X denote a discrete random variable with pmf of px (1) 75 and pr (2)...
Let X denote a discrete random variable with pmf of px (1) 75 and pr (2) = .25. When the random variable X is transmitted, the
Let Xn, -inf to +inf be a discrete-time zero-mean white noise process, i.e., μx[n] = 0, Rx[n] =δ[...
Let Xn, -inf to +inf be a discrete-time zero-mean white noise
process, i.e., μx[n] = 0, Rx[n] =δ[n]. The process is filtered
using an LTI system with impulse response
h[n] =αδ[n] + βδ[n−1].
Find α and β such that the output process Yn has autocorrelation
function Ry[n] =δ[n+1] + 2δ[n] +δ[n−1].
5) (3 points) Let Xn, -o0 K n oo, be a discrete-time zero-mean white noise process, i.e, ,1z[n]-(), Rx [n] S[n]. The process is filtered using an LTI system...
Problem 1. Let X be a discrete random variable with values -2,0,1,5 urith pmf (a) Verify...
Problem 1. Let X be a discrete random variable with values -2,0,1,5 urith pmf (a) Verify that the probabilities do define a pmf (probability mass function) ( b) Compute the mean of X , i.e., μ -E(X) (c) Compute the standard deviation of X, i.e., σ- Nar(X)
Problem 4 Let X be the following discrete random variable: P(X-1) = P(X = 0) =...
Problem 4 Let X be the following discrete random variable: P(X-1) = P(X = 0) = P(x-1) Let Y-X2. Show that cov(X, Y) 0, but X and Y are not independent random variable.
Let x be a discrete random variable with PR mass function f(x)=2(1/3)^x, x=1,2,3.. A) Compute Mx(t)...
Let x be a discrete random variable with PR mass function f(x)=2(1/3)^x, x=1,2,3.. A) Compute Mx(t) B) Compute M'1=EX, M'2=EX^2
Problem 3. Let X be a discrete random variable that takes values in N. Show that...
Problem 3. Let X be a discrete random variable that takes values in N. Show that if X is memory-free then it must be the case that X Geo(p) for some p. (Hint a useful first step might be to show that P(X > t)= P(X > 1)' for all t E N.)
Problem 3. Let X be a discrete random variable that takes values in N. Show that if X is memory-free then it must be the case that...
1 Let X be a discrete random variable. (a) Show that if X has a finite...
1 Let X be a discrete random variable. (a) Show that if X has a finite mean μ. then EX-ix-0. (b) Show that if X has a finite variance, then its mean is necessarily finite 2 Let X and Y be random variables with finite mean. Show that, if X and Y are independent, then 3 Let Y have mean μ and finite variance σ2 (a) Use calculus to show that μ is the best predictor of Y under quadratic...
Problem 4 Let X be a discrete random variable with probability mass function fx(x), and let t be a function. Define Y = t(X): that is, Y is the randon variable obtained by applying the function t to the value of X Transforming a random variable in this way is frequently done in statistics. In what follows, let R(X) denote the possible values of X and let R(Y) denote the possible values of To compute E[Y], we could irst find...
Let X be a discrete random variable, and let Y X (a) Assume that the PMF of X is Ka2 0 if x- -3, -2,-1,0,1,2,3 otherwise, where K is a suitable constant. Determine the value of K. (b) For the PMF of X given in part (a) calculate the PMF of Y (c) Give a general formula for the PMF of Y in terms of the PMF of X
Let X denote a discrete random variable with pmf of px (1) 75 and pr (2) = .25. When the random variable X is transmitted, the
Let Xn, -inf to +inf be a discrete-time zero-mean white noise
process, i.e., μx[n] = 0, Rx[n] =δ[n]. The process is filtered
using an LTI system with impulse response
h[n] =αδ[n] + βδ[n−1].
Find α and β such that the output process Yn has autocorrelation
function Ry[n] =δ[n+1] + 2δ[n] +δ[n−1].
5) (3 points) Let Xn, -o0 K n oo, be a discrete-time zero-mean white noise process, i.e, ,1z[n]-(), Rx [n] S[n]. The process is filtered using an LTI system...
Problem 1. Let X be a discrete random variable with values -2,0,1,5 urith pmf (a) Verify that the probabilities do define a pmf (probability mass function) ( b) Compute the mean of X , i.e., μ -E(X) (c) Compute the standard deviation of X, i.e., σ- Nar(X)
Problem 4 Let X be the following discrete random variable: P(X-1) = P(X = 0) = P(x-1) Let Y-X2. Show that cov(X, Y) 0, but X and Y are not independent random variable.
Problem 3. Let X be a discrete random variable that takes values in N. Show that if X is memory-free then it must be the case that X Geo(p) for some p. (Hint a useful first step might be to show that P(X > t)= P(X > 1)' for all t E N.)
Problem 3. Let X be a discrete random variable that takes values in N. Show that if X is memory-free then it must be the case that...
1 Let X be a discrete random variable. (a) Show that if X has a finite mean μ. then EX-ix-0. (b) Show that if X has a finite variance, then its mean is necessarily finite 2 Let X and Y be random variables with finite mean. Show that, if X and Y are independent, then 3 Let Y have mean μ and finite variance σ2 (a) Use calculus to show that μ is the best predictor of Y under quadratic...
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