a. The table would be as follows:
X\Y | 0 | 1 | 2 | P(X = x)
---|---------|---------|---------|---------
0 | 3/9 | 2/9 | 0/9 | 5/9
1 | 0/9 | 4/9 | 2/9 | 6/9
2 | 0/9 | 0/9 | 0/9 | 0/9
**P(Y = y)** | **3/9** | **6/9** | **2/9** | **1**
b. The join cumulative probability distribution for all values would be 1.
c. The marginal probability for fX(x) is 1/3 and for fY(y) is (1/3) (y + 2).
a. **Joint Probability Mass Function (PMF) Table:**
X\Y | 0 | 1 | 2 | P(X = x)
---|---------|---------|---------|---------
0 | 3/9 | 2/9 | 0/9 | 5/9
1 | 0/9 | 4/9 | 2/9 | 6/9
2 | 0/9 | 0/9 | 0/9 | 0/9
**P(Y = y)** | **3/9** | **6/9** | **2/9** | **1**
Here, P(X = x, Y = y) represents the probability of selecting x Republicans and y Democrats.
b. **Joint Cumulative Probability Distribution Function (CDF):**
F(0,0) = P(X ≤ 0, Y ≤ 0) = P(X = 0, Y = 0) = 3/9
F(1,0) = P(X ≤ 1, Y ≤ 0) = P(X = 0, Y = 0) + P(X = 1, Y = 0) = 3/9
F(2,0) = P(X ≤ 2, Y ≤ 0) = P(X = 0, Y = 0) + P(X = 1, Y = 0) + P(X = 2, Y = 0) = 3/9
F(0,1) = P(X ≤ 0, Y ≤ 1) = P(X = 0, Y = 0) + P(X = 0, Y = 1) = 5/9
F(1,1) = P(X ≤ 1, Y ≤ 1) = P(X = 0, Y = 0) + P(X = 1, Y = 0) + P(X = 1, Y = 1) = 9/9
F(2,1) = P(X ≤ 2, Y ≤ 1) = P(X = 0, Y = 0) + P(X = 1, Y = 0) + P(X = 2, Y = 0) + P(X = 2, Y = 1) = 9/9
F(0,2) = P(X ≤ 0, Y ≤ 2) = P(X = 0, Y = 0) + P(X = 0, Y = 1) + P(X = 0, Y = 2) = 1
F(1,2) = P(X ≤ 1, Y ≤ 2) = P(X = 0, Y = 0) + P(X = 1, Y = 0) + P(X = 1, Y = 1) + P(X = 1, Y = 2) = 1
F(2,2) = P(X ≤ 2, Y ≤ 2) = P(X = 0, Y = 0) + P(X = 1, Y = 0) + P(X = 2, Y = 0) + P(X = 2, Y = 1) + P(X = 2, Y = 2) = 1
c. **Marginal Probability Mass Functions:**
f_X(x) = P(X = x)
= ∑_y P(X = x, Y = y) (Sum over all possible values of Y)
= ∑_y P(X = x and Y = y) (Since P(X = x, Y = y) = P(X = x and Y = y))
= ∑_y (3x/9) (From the joint PMF table)
= (3x/9) ∑_y 1 (Since the sum over all possible values of Y is 1)
= (3x/9) ⋅ 1 (Sum over all possible values of Y is 1)
= 1/3
Similarly,
f_Y(y) = P(Y = y)
= (3y + 6)/9
= (1/3) (y + 2)
The question probable may be:
From a group of 3 republicans, 4 democrats, and 2 independents, a committee of two people is selected at random.
Let X be the number of republicans chosen.
Let Y be the number of democrats chosen.
a. Make a table showing the joint probability mass function.
b. Find the joint cumulative probability distribution function for all values of ( a, b)
c. Find the marginal probability mass functions: fX (x) and fY(y)