1. The discrete random variables X and Y have known joint probability mass function [c x=0,1,2,L,9, y = 0, 1, 2,L,9, y≥x с where c = 1/55. 0 otherwise Py(x, y) given by Pxy(x, y) =< Define the event B = {X² +Y² ≤25}. a) Determine the conditional PMF P(x, y/B) of the random variables X and Y. b) Determine the conditional marginal PMFs P(x/B) and P(y/B) of the random variables X and Y. c) Determine the conditional correlation E{XY/B} of the random variables X and Y. (Take advantage of the solutions to Prob. 1 in Homework #8.) 2

1. The discrete random variables X and Y have known joint probability mass function [c x=0,1,2,L,9, y = 0, 1, 2,L,9, y≥x с where c = 1/55. 0 otherwise Py(x, y) given by Pxy(x, y) =< Define the event B = {X² +Y² ≤25}. a) Determine the conditional PMF P(x, y/B) of the random variables X and Y. b) Determine the conditional marginal PMFs P(x/B) and P(y/B) of the random variables X and Y. c) Determine the conditional correlation E{XY/B} of the random variables X and Y. (Take advantage of the solutions to Prob. 1 in Homework #8.) 2

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y): - 30 -50x²-50y² +80xy for -
Suppose that the random variables X,Y, and Z have the joint probability density function f(x,y,z) = 8xyz for 0<x<1, 0<y<1, and 0<z<1. Determine P(X<0.7).
Let the stochastic process {Xt} be defined as Zt ; if t is even (Z2t-1 -1)=21/2; if t is uneven, where {Zt} is identically and independently distributed as Zt is N(0, 1). Show that {Xt} is WN(0, 1), but not IID (0,1).
2)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the maximum likelihood estimator (MLE) of parameter θ.
Suppose that {Xn, n ≥ 1} is a sequence of discrete random variables with distribution functionP(Xn =i/n) = 2i/(n(n + 1)), for i = 1, 2, ..., n.Let X be a continuous random variable with density functionfX(x) = 2x for 0 < x < 1,0 , otherwise(a) Compute P(X ≤ x) for 0 < x < 1;(b) Compute P(Xn ≤m/n) for some integer m such that 1 ≤ m ≤ n;(c) Compute P(Xn ≤ x) for 0 < x < 1;(d) Prove that Xnd→ X.
Let X1, X2, X3, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let Sn = X1 + · · · + Xn. (a) Show that GXi(s) = es−1.(b) Deduce from part (a) that GSn(s) = ens−n.
(a) Let Y1 , Y2 , · · · , Yn be independent and identically distributed random variables. Show if Y1 < Y2 is independent of the event Y1 < Y3.(b) Let Y1 , Y2 , · · · , Y∞ be dependent random variables with following probability density function; Yn+1 =3Yn with probability 1/3Yn+1 =Yn/3 with probability 2/3 for n ∈ [0, ∞) where Y0 = 1. Compute E[Y1] and E[Y2].
4 (b) An insurance company provides customers with both auto and home insurance policies. For a particular customer, Χ is the deduction on his or her auto policy and Y is the deduction on the home policy. Possible values of Χ are K100 and K250, and for Y are K0, K100 and K200. The joint probability density function for ( ) ,YΧ is given by the following table: Χ Y K100 K250 K0 0.20 0.05 K100 0.10 0.15 K200 0.20 0.30 iv. If we look only at those insurance customers selecting the lowest auto mobile insurance deduction (K100), what is the probability that a randomly selected\ customer will also select the lowest home deduction (K0). v. Compute the correlation coefficient of Χ and Y
If a random variable X has a discrete uniform distribution. fx(x)=1/k for x=1,2,..,k;0 otherwise. Derive P.G.F of X and compute E(2x+1)
The joint probability function of two discrete random variables X and Y is given by Ax,y) = c(2x+y), where x and y can assume all integers such that 0< x
The joint probability mass function of XX and YY is given by p(1,1)=0.45p(2,1)=0.05p(3,1)=0.05p(1,2)=0.05p(2,2)=0.1p(3,2)=0.1p(1,3)=0.05p(2,3)=0.05p(3,3)=0.1p(1,1)=0.45p(1,2)=0.05p(1,3)=0.05p(2,1)=0.05p(2,2)=0.1p(2,3)=0.05p(3,1)=0.05p(3,2)=0.1p(3,3)=0.1 Compute the following probabilities:P(X+Y>3)=P(X+Y>3)=P(XY=2)=P(XY=2)=P(XY>1)=P(XY>1)=
Suppose the joint probability density of X and Y is fX,Y (x, y) = 3y 2 with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and zero everywhere else. 1. Compute E[X|Y = y]. 2. Compute E[X3 + X|X < .5]