To find the joint probability distribution function for X and Y, we need to determine the probability of each possible outcome for X and Y.
There are a total of 8 refills in the box: 3 blue, 2 red, and 3 green. So the total number of ways to choose 2 refills out of 8 is given by the binomial coefficient C(8,2) = 28.
Now, let's consider the possible outcomes for X and Y:
1. X = 0, Y = 0: The probability of selecting 2 green refills is C(3,2)/C(8,2) = 3/28.
2. X = 1, Y = 0: The probability of selecting 1 blue and 1 green refill is C(3,1) * C(3,1)/C(8,2) = 9/28.
3. X = 0, Y = 1: The probability of selecting 1 red and 1 green refill is C(2,1) * C(3,1)/C(8,2) = 6/28.
4. X = 2, Y = 0: The probability of selecting 2 blue refills is C(3,2)/C(8,2) = 3/28.
5. X = 1, Y = 1: The probability of selecting 1 blue and 1 red refill is C(3,1) * C(2,1)/C(8,2) = 6/28.
6. X = 0, Y = 2: The probability of selecting 2 red refills is C(2,2)/C(8,2) = 1/28.
Therefore, the joint probability distribution function for X and Y is as follows:
P(X=0, Y=0) = 3/28
P(X=1, Y=0) = 9/28
P(X=0, Y=1) = 6/28
P(X=2, Y=0) = 3/28
P(X=1, Y=1) = 6/28
P(X=0, Y=2) = 1/28
These probabilities add up to 1, as they should for a valid joint probability distribution function.
Two refills for a ball point pen are selected at random from a box that contains 3 blue, 2
red and 3 green refills. Let
X
be the number of the blue refills and
Y
be the number of
red refills. Find the joint probability distribution function
X
and
Y .
1 answer
1 answer