The probability distribution of a random variable X, the number of defects per 10 meters of a fabric is given by | x | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X = x) | 0.45 | 0.35 | 0.15 | 0.03 | 0.02 | Find the variance of X. **OR** For the following probability density function (p.d.f.) of X, find: (i) P(X < 1), (ii) P(|X| < 1) if $f(x) = \dfrac{x^2}{18}$, $-3 < x < 3$ $= 0$, otherwise

> > [!TLDR] > Part 1: $\mathrm{Var}(X)=E(X^2)-[E(X)]^2$ from the table. Part 2 (OR): integrate the pdf over the given ranges. > > **Part 1 — Variance from the given table:** > > | x | 0 | 1 | 2 | 3 | 4 | > |---|---|---|---|---|---| > | P(X=x) | 0.45 | 0.35 | 0.15 | 0.03 | 0.02 | > > $$E(X) = \sum xP(x) = 0(.45)+1(.35)+2(.15)+3(.03)+4(.02) = 0+.35+.30+.09+.08 = 0.82$$ > > $$E(X^2) = \sum x^2P(x) = 0(.45)+1(.35)+4(.15)+9(.03)+16(.02) = 0+.35+.60+.27+.32 = 1.54$$ > > $$\mathrm{Var}(X) = E(X^2)-[E(X)]^2 = 1.54 - (0.82)^2 = 1.54-0.6724 = 0.8676$$ > > --- > > **Part 2 (OR)** — pdf $f(x)=\dfrac{x^2}{18}$, $-3 …

The probability distribution of a random variable X, the number of defects per 10 meters of a fabric is given

Question 75 of 88

The probability distribution of a random variable X, the number of defects per 10 meters of a fabric is given by

x	0	1	2	3	4
P(X = x)	0.45	0.35	0.15	0.03	0.02

Find the variance of X.

For the following probability density function (p.d.f.) of X, find: (i) P(X < 1), (ii) P(|X| < 1)

if $f(x) = \dfrac{x^2}{18}$ , $-3 < x < 3$

$= 0$ , otherwise

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2019Subjective· 3mImportance★★★★★

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Part 1: $\mathrm{Var}(X)=E(X^2)-[E(X)]^2$ from the table. Part 2 (OR): integrate the pdf over the given ranges.

Part 1 — Variance from the given table:

x	0	1	2	3	4
P(X=x)	0.45	0.35	0.15	0.03	0.02

$E(X) = \sum xP(x) = 0(.45)+1(.35)+2(.15)+3(.03)+4(.02) = 0+.35+.30+.09+.08 = 0.82$

$E(X^2) = \sum x^2P(x) = 0(.45)+1(.35)+4(.15)+9(.03)+16(.02) = 0+.35+.60+.27+.32 = 1.54$

$\mathrm{Var}(X) = E(X^2)-[E(X)]^2 = 1.54 - (0.82)^2 = 1.54-0.6724 = 0.8676$

Part 2 (OR) — pdf $f(x)=\dfrac{x^2}{18}$ , $-3<x<3$ :

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