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Mean and variance of a random variable — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMean and variance of a random variable1 Mark
MCQ
A random variable has the following probability distribution:
$X = x_i$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
$P(X = X_i)$ $0$ $2p$ $2p$ $3p$ $p^2$ $2p^2$ $7p^2$ $2p$
  • ✓
    $\frac{1}{10}$
  • B
    $-1$
  • C
    $-\frac{1}{10}$
  • D
    $\frac{1}{5}$
✓

Answer

Correct option: A.
$\frac{1}{10}$
We know that the sum of probabilities in a probability distribution is always $1.$
$\therefore P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) +P(X = 6) + P(X = 7) +P(X = 8) = 1$
$\Rightarrow 0 + 2p + 2p + 3p + p^2 + 2p^2 + 7p^2 + 2p = 1$
$\Rightarrow 10p^{2 }+ 9p - 1 = 0$
$\Rightarrow (10p - 1)(p + 1) = 0$
$\Rightarrow\text{p}=\frac{1}{10}\text{ or }-1 ($Negleting $-1$ as the value of the probability cannot be negative$)$

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