Given that x and y are independent.
Hence f(x,y) = f(x) g(y)
W = x+y This implies that x =x and y = w-x
Hence x = x and y = w-x, for x = 0,1,2......w
Hence P(w) = P(x =x, y = w-x) = f(x) g(w-x), x = 0,1,2,....w
, w=0, 1, 2, ...
Hence proved
5.4-9. Let X and Y, with respective pmfs f(x) and g(y), be independent discrete ran- dom...
ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that the function p: R2 R defined by p(r, y) px(x)pr(y) is a joint PMF by verifying that it satisfies properties (a)-(c) of Proposition 6.1 on page 262. Hint: A subset of a countable set is countable CHAPTER SIX Joindy Discrete Random Variables 6.2 Joint and marginal PMFs of the discrete random variables x numher of bedrooms and momber of bwthrooms of a...
Proposition 6.10 Independent Discrete Random Variables: Bivariate Case Let X andY be two discrete random variables defined on the same sample space. Then X and Y are independent if and only if pxy(x,y) = px(x)py(y), for all x , y ER. (6.19) In words, two discrete random variables are independent if and only if their joint equals the product of their marginal PMFs. Proposition 6.11 Independence and Conditional Distributions Discrete random variables X and Y are independent if and only...
Let X and Y be independent random variables. Random variable X has a discrete uniform distribution over the set {1, 3} and Y has a discrete uniform distribution over the set {1, 2, 3}. Let V = X + Y and W = X − Y . (a) Find the PMFs for V and W. (b) Find mV and (c) Find E[V |W >0].
I need help by solving this Problems 133 5.3 Let X and Y have the following joint PMF 0.1 0.1 0.1 0.1 0 0.0 0.1 0.2 0.3 o-一ㄒㄧㄒㄧ丁 a. What are the marginal PMFs of X and Y? b. What are the conditional probabilities (computed directly) of X given Y and Y giw X (compute them directly)? c. What are the conditional probabilities of Y given X from the conditional probabilit of X given Y using Bayes theorem? Using the...
8. We say that two discrete random variables X and Y , are independent when P(X = a, Y = b) = P(X = a)P(Y = b) for all a and b in the corresponding sample spaces. Let Xị and X, be independent Poisson random variables with parameters l1 = 3 and dy = 2 respectively. Find the probability of the event that X1 + X2 = 3. Hint: Since {X1 + X2 = 3} = {X} = 0, X2...
33. Let X and Y be independent exponential random variables with respective rates λ and μ. (a) Argue that, conditional on X> Y, the random variables min(X, Y) and X -Y are independent. (b) Use part (a) to conclude that for any positive constant c E[min(X, Y)IX > Y + c] = E[min(X, Y)|X > Y] = E[min(X, Y)] = λ+p (c) Give a verbal explanation of why min(X, Y) and X - Y are (unconditionally) independent. 33. Let X...
6 X and Y are two discrete random variables with the following PMF. IN IN IA. a. | Find the marginal pmf's for X and Y. b. Draw the joint CD c. Calculate the probability of the events: A-(X>0), B (xeY), and C-X Y for the 3 pt 3 pt. indicated PMF t. Are X, Y independent? Prove. 2 pt. t.
Let X and Y denote independent random variables with respective probability density functions, f(x) = 2x, 0<x<1 (zero otherwise), and g(y) = 3y2, 0<y<1 (zero otherwise). Let U = min(X,Y), and V = max(X,Y). Find the joint pdf of U and V.
. Jan wins, ue OTCS LICO. 6.127 Consider two identical electrical components. Let X and Y denote the respective lifetimes of the two components observed at discrete time units (e.g.,every hour). Assume that the joint PMF of X and Y is px.y(x, y) = p (1 - p)*+-2 if x, y e N, and px.Y(x, y) 0 otherwise, where O <p< l. Use the FPF for two discrete random variables to determine the probability that one of the components lasts...
Let X and Y be two independent random variables with X =d R(0, 2) and Y =d exp(1). (a) Use the convolution formula to calculate the probability density function of W =X+Y. (b) Derive the probability density function of U = XY .