Joint Pdfs
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
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 fX,Y(x, y).
fX,Y(x, y)=1/4П for x2 + y2 ≤ 4
-100
For each of the following joint pdfs, find FX,Y (𝑢, 𝑣). fX,Y (𝑥, 𝑦) = 3/2y2 , 0 ≤ x ≤ 2, 0 ≤ y ≤ 1.
FX,Y (𝑢, 𝑣)=1/2𝑢𝑣3, 0 ≤ x ≤ 2, 0 ≤ y ≤ 1
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
!!!BOOM!!!
-100
Find fY(y) if fX,Y(x, y) = 2e−x e−y for (x, y) defined over the shaded region pictured.
fY(y)= 2(e−y - e−2y)
Find the joint pdf associated with two random variables X and Y whose joint CDF is FX,Y(x, y) = (1 − e−λy )(1 − e−λx ), x > 0, y > 0.
FX,Y(x, y)=λe-λ(x+y) , x > 0, y > 0
Find P(X < 2Y) if fX,Y(x, y) = x + y for X and Y each defined over the unit interval
P(X < 2Y)=19/24
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
Suppose that fX,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.
fX(x)=3(1-x)2
For x<0 or x>1, fX(x)=0
~~~BONUS~~~
+100