Question
Two cards are selected at random from a box which contains five cards numbered $1, 1, 2, 2,$ and $3$. Let X denote the sum and Y the maximum of the two numbers drawn. Find the probability distribution, mean and variance of X and Y.
| x: | 2 | 3 | 4 | 5 |
| P(x): | 0.1 | 0.4 | 0.3 | 0.2 |
| $x_i$ | $p_i$ | $x_ip_i$ | $x_i^2p_i$ |
| 2 | 0.1 | 0.1 | 0.4 |
| 3 | 0.4 | 1.2 | 3.6 |
| 4 | 0.3 | 1.2 | 4.8 |
| 5 | 0.2 | 1.0 | 5.0 |
| $\sum\text{xp}=3.6$ | $\sum\text{x}^2\text{p}=13.8$ |
|
x:
|
1
|
2
|
3
|
|
p(x):
|
0.1
|
0.5
|
0.4
|
| $y_i$ | $p_i$ | $y_ip_i$ | $y_i^2p_i$ |
| 1 | 0.1 | 0.1 | 0.1 |
| 2 | 0.5 | 1.0 | 2.0 |
| 3 | 0.4 | 1.2 | 3.6 |
| $\sum\text{xp}=2.3$ | $\sum\text{x}^2\text{p}=5.7$ |
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