Let X denote the number of hours a person watches television during a randomly selected day. The probability that X can take the values $x_i$ has the following form, where k is some unknown constant.
$P(X = x;) =\left\{\begin{matrix}0.2,&if\,x_i=0\\Kx_i,& if\, x_i = 1\, or\, 2\\k (5-x_i),&if\, x_i = 3\\0,&otherwise\end{matrix}\right.$
Calculate mathematical expectation.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1.76
From the given information, we find that the probability distribution of X is
|
X |
0 |
1 |
2 |
3 |
|
P(X) |
0.2 |
k |
2k |
2k |
We know that $Σp_i = 1$
$⇒ 0.2+k+2k + 2k = 1$
$⇒ 5k=0.8⇒k=\frac{4}{25}$
We construct the following table:
|
$x_i$ |
$p_i$ |
$p_ix_i$ |
$p_i{x_i}^2$ |
|
0 |
0.2 |
0 |
0 |
|
1 |
$\frac{4}{25}$ |
$\frac{4}{25}$ |
$\frac{4}{25}$ |
|
2 |
$\frac{8}{25}$ |
$\frac{16}{25}$ |
$\frac{32}{25}$ |
|
3 |
$\frac{8}{25}$ |
$\frac{24}{25}$ |
$\frac{72}{25}$ |
|
Total |
|
$\frac{44}{25}$ |
$\frac{108}{25}$ |
$E(X) = Σp_ix_i=\frac{44}{25}= 1.76$