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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES5 Marks
Question
Three numbers are selected at random (without replacement) from first six positive integers. If X denotes the smallest of the three numbers obtained, find the probability distribution of X. Also find the mean and variance of the distribution.
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Answer

Total number of ways $^6\text{C}_3=20$
$\text{X:}$ 1 2 3 4
$\text{P(X):}$ $\frac{10}{20}$ $\frac{6}{20}$ $\frac{3}{20}$ $\frac{1}{20}$
$\text{XP(X):}$ $\frac{10}{20}$ $\frac{12}{20}$ $\frac{9}{20}$ $\frac{4}{20}$
$\text{X}^{2}\text{P(X)}:$ $\frac{10}{20}$ $\frac{24}{20}$ $\frac{27}{20}$ $\frac{16}{20}$
$\text{Mean} = \sum \text{XP(X)} =\frac{35}{20} = \frac{7}{4}$
$\text{Variance} = \sum \text{X}^{2}\text{P(X)} - \bigg[\sum\text{XP(X)}\bigg]^{2}= \frac{77}{20} - \frac{49}{16} = \frac{63}{80}$

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