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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