Question
Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X.
✓
Answer
The first five positive integers are 1, 2, 3, 4, 5 we select two positive numbers in 5 × 4 = 20 ways.
Out of these two no. are selected at random.
let X denote larger of the two no.
X can be 2, 3, 4 or 5.
P(X = 2) = P(larger no. is 2) = {(1, 2) and (2, 1)}
$=\frac{2}{30}$
$\text{P}(\text{X}=3)=\frac{4}{30}$
$\text{P}(\text{X}=4)=\frac{6}{30}$
$\text{P}(\text{X}=5)=\frac{8}{30}$
$\text{Mean}=\text{E}(\text{X})=2\times\frac{2}{30}+3\times\frac{4}{30}+4\times\frac{6}{30}+5\times\frac{8}{3 0}$
$=\frac{4+12+24+40}{30}$
$=\frac{80}{30}$
$\text{Variance}=2^2\times\frac{2}{30}+3^2\times\frac{4}{30}+4^2\times\frac{6}{30}+5^2\times\frac{8}{30}$
$=\frac{8+36+96+200}{30}$
$=\frac{340}{30}=\frac{34}{3}$
Out of these two no. are selected at random.
let X denote larger of the two no.
X can be 2, 3, 4 or 5.
P(X = 2) = P(larger no. is 2) = {(1, 2) and (2, 1)}
$=\frac{2}{30}$
$\text{P}(\text{X}=3)=\frac{4}{30}$
$\text{P}(\text{X}=4)=\frac{6}{30}$
$\text{P}(\text{X}=5)=\frac{8}{30}$
$\text{Mean}=\text{E}(\text{X})=2\times\frac{2}{30}+3\times\frac{4}{30}+4\times\frac{6}{30}+5\times\frac{8}{3 0}$
$=\frac{4+12+24+40}{30}$
$=\frac{80}{30}$
$\text{Variance}=2^2\times\frac{2}{30}+3^2\times\frac{4}{30}+4^2\times\frac{6}{30}+5^2\times\frac{8}{30}$
$=\frac{8+36+96+200}{30}$
$=\frac{340}{30}=\frac{34}{3}$
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