The probability distribution of a random variable x is given as u… - Vidyadip

Question

The probability distribution of a random variable x is given as under:
$\text{P}(\text{X}=\text{x})=\begin{cases}\text{k}\text{x}^2 & \text{for}\text{ x}= 1,2,3\\2\text{kx} & \text{for}\text{ x } =4,5,6\\0&\text{otherwise} \end{cases}$
where k is a constant. Calculate

$\text{E}(\text{X})$
$\text{E}(3\text{X}^2)$
$\text{P}(\text{X}\geq4)$

✓

Answer

Given condition can be represented as below:

X	1	2	3	4	5	6	Otherwise
P(X)	k	4k	9k	8k	10k	12k	0

We know that, $\sum\text{P}_{\text{i}}=1$

$\Rightarrow44\text{k}=1$

$\Rightarrow\text{k}=\frac{1}{44}$

$\therefore\sum\text{XP}(\text{X})=\text{k}+8\text{k}+27\text{k}+32\text{k}+50\text{k}+72\text{k}+0$

$=190\text{k}=190\times\frac{1}{44}=\frac{95}{22}$

So, $\text{E}(\text{X})=\sum\text{XP}(\text{X})=\frac{95}{22}=4.32$
Also, $\text{E}(\text{X}^2)=\sum\text{X}^2\text{P}(\text{X})=\text{k}+16\text{k}+128\text{k}+250\text{k}+432$

$=908\text{k}=908\times\frac{1}{44}$ $\Big[\because\text{k}=\frac{1}{44}\Big]$

$=20.636=20.64$ (approx)

$\therefore\text{E}(3\text{X}^2)=3\text{E}(\text{X}^2)=3\times20.64$

$=61.92=61.9$

$\text{P}(\text{X}\geq4)=\text{P}(\text{X}=4)+\text{P}(\text{X}=5)+\text{P}(\text{X}=6)$

$=8\text{k}+10\text{k}+12\text{k}=30\text{k}$

$=30\cdot\frac{1}{44}=\frac{15}{22}$

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