Enter Title

Joint

100

1. A class consists of 21 students: 7 students are from Group 1, 9 students are from Group 2, and 5 students are from Group 3. A random sample of size 3 is selected without replacement. Let X denote the number of students selected from Group 2 and Y denote the number of students selected from Group 3. Write a formula for the joint pdf of X and Y.

P(X=x, Y=y)=0

200

Two fair dice are rolled once. Let X denote the number of 3’s that appear, and Y denote the number of 4’s that appear. Define a new random variable Z = X + Y, where Z represents the total number of dice showing either 3 or 4. Find the probability P(Z ≤ 1).

P(Z≤1)=8/9

300

Find P(X < 2Y) if f_XY(x, y) = x + y for X and Y each defined over the unit interval.

P(X<2Y)=19/24

Geometric Probability

100

A point is chosen at random from the interior of a circle whose equation is x² + y² ≤ 4. Let the random variables X and Y denote the x-and y-coordinates of the sampled point. Find f_XY(x, y).

f_XY(x,y)=1/4п for x²+y²≤4

200

A point is chosen at random from the interior of a circle with radius R. Let the random variables X and Y denote the x-and y-coordinates of the selected point. Find the probability that the point lies in the upper half of the ring between circles of radius R/2 and R.

P=3/8

300

!!!BOOM!!!

-100

Marginal Pdfs

100

Find fY(y) if fX,Y(x, y) = 2e −x e −y for (x, y) defined over the shaded region pictured.

f_Y(y)=2(e^-y-e^-2y)

200

Suppose that f_X,Y(x, y) = 6(1 − x − y) for xand ydefined over the unit square, subject to the restriction 0 ≤ x + y ≤ 1. Find the marginal pdf of X.

f_X(x)=3(1-x)²

300

!!!BOOM!!!

-100

Joint Cdfs

100

For each of the following joint pdfs, find F_X,Y (𝑢, 𝑣). f_X,Y (𝑥, 𝑦) = 3/2y² , 0 ≤ x ≤ 2, 0 ≤ y ≤ 1.  

F_X,Y(𝑥, 𝑦)=1/2𝑢𝑣³,  0 ≤ x ≤ 2, 0 ≤ y ≤ 1.  

200

Find the joint pdf associated with two random variables X and Y whose joint CDF is F_X,Y(x, y) = (1 − e^−λy )(1 − e^−λx ), x > 0, y > 0.

f_X,Y (𝑥, 𝑦) = λe^−λx*λe^−λy=λ²e^-−λ(x+y), x > 0, y > 0.

300

!!!BONUS!!!

+100