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Probability Distributions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsProbability Distributions2 Marks
MCQ
A random variable X takes values 1,2,3,4 with probabilities $\frac{1}{6}, \frac{1}{3}, \frac{1}{3}, \frac{1}{6}$ respectively, then its mean and variance are respectively
  • ✓
    $\frac{5}{2}, \frac{11}{12}$
  • B
    $\frac{5}{2}, \frac{11}{16}$
  • C
    $\frac{5}{3}, \frac{11}{16}$
  • D
    $\frac{5}{3}, \frac{11}{12}$
✓

Answer

Correct option: A.
$\frac{5}{2}, \frac{11}{12}$
(A)
$\text {Mean}=\mathrm{E}(\mathrm{X})=\sum x_{\mathrm{i}} \cdot \mathrm{P}\left(x_{\mathrm{i}}\right)$
$ =\frac{1}{6}(1)+\frac{1}{3}(2)+\frac{1}{3}(3)+\frac{1}{6}(4) $
$ =\frac{1+4+6+4}{6} $
$ =\frac{15}{6}=\frac{5}{2} $
$ \text { Variance }=\mathrm{E}\left(\mathrm{X}^2\right)-[\mathrm{E}(\mathrm{X})]^2 $
$ =\frac{1}{6}(1)^2+\frac{1}{3}(2)^2+\frac{1}{3}(3)^2+\frac{1}{6}(4)^2-\left(\frac{5}{2}\right)^2 $
$ =\frac{1}{6}+\frac{4}{3}+\frac{9}{3}+\frac{16}{6}-\frac{25}{4} $
$ =\frac{2+16+36+32-75}{12} $
$ =\frac{86-75}{12} $
$ =\frac{11}{12}$

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