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JEE Main 20245 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
5 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Statistics

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Statistics

From a lot of 10 items, which include 3 defective items, a sample...

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable X\mathrm{X} denote the number of defective items in the sample. If the variance of X\mathrm{X} is σ2\sigma^2, then 96σ296 \sigma^2 is equal to ______

Correct answer

0

Step-by-step solution

X=denotesnumberofdefectivedenotes number of defectivex0123P(x)715512512112x120149Pix1205122012912pixi05121012312\begin{array}{|c|c|c|c|c|} \hline \mathrm{x} & 0 & 1 & 2 & 3 \\ \hline \mathrm{P}(\mathrm{x}) & \frac{7}{15} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\ \hline \mathrm{x}_1{ }^2 & 0 & 1 & 4 & 9 \\ \hline \mathrm{P}_{\mathrm{i}} \mathrm{x}_1{ }^2 & 0 & \frac{5}{12} & \frac{20}{12} & \frac{9}{12} \\ \hline \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} & 0 & \frac{5}{12} & \frac{10}{12} & \frac{3}{12} \\ \hline \end{array}μ=Σpixi=1812\mu=\Sigma \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}=\frac{18}{12}Σpix12=3412σ2=Σpix12−(μ)2\begin{aligned} & \Sigma \mathrm{p}_{\mathrm{i}} \mathrm{x}_1^2=\frac{34}{12} \\ & \sigma^2=\Sigma \mathrm{p}_{\mathrm{i}} \mathrm{x}_1^2-(\mu)^2 \end{aligned}=3412−(1812)2=176−9434−2712=71296σ2=96×712=56\begin{aligned} & =\frac{34}{12}-\left(\frac{18}{12}\right)^2=\frac{17}{6}-\frac{9}{4} \\ & \frac{34-27}{12}=\frac{7}{12} \\ & 96 \sigma^2=96 \times \frac{7}{12}=56\end{aligned}$$
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