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Probability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSProbability1 Mark
Question
If $X$ is a random variable with probability distribution as given below$:$
$X = x_i$​​​​​​​ $0$ $1$ $2$ $3$
$P(X = X_i)$ $k$ $3k$ $3k$ $k$
The value of $k$ and its variance are$:$
✓

Answer

$\sum\limits_0^3\text{P}(\text{x})=1$
$\text{k}+3\text{k}+3\text{k}+\text{k}=1$
$\text{k}=\frac{1}{8}$
$\text{x}$ $\text{P}(\text{x})$ $\text{x}\text{P}(\text{x})$ $\text{x}^2\text{P}(\text{x})$
$0$ $\frac{1}{8}$ $0$ $0$
$1$ $\frac{3}{8}$ $\frac{3}{8}$ $\frac{3}{8}$
$2$ $\frac{3}{8}$ $\frac{6}{8}$ $\frac{12}{8}$
$3$ $\frac{1}{8}$ $\frac{3}{8}$ $\frac{9}{8}$
$\text{Total}$   $\text{E(x)}=\frac{12}{8}=1.5$ $\text{E}(\text{x}^2)=3$
$\text{V(x)}=\text{E}(\text{x}^2)-[\text{E}(\text{x})^2]$
$\text{V(x)}=3-(1.5)^2$
$\text{V(x)}=0.75$
$=\frac{3}{4}$

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