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Question Bank [2022] — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Question Bank [2022]4 Marks
Question
If a $r.v. x.$ has $p.d.f f(x)=\left\{\begin{array}{l}\frac{c}{x}, 10 \\ 0, \text { otherwise }\end{array}\right.$ Find $c, E(X), $ and $\operatorname{Var}(X)$. Also Find $F(x)$.
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Answer

$a.$ Given that $f(x)$ represents $p.d.f.$ of $r.v. X$
$ \therefore \int_1^3 f(x) \cdot d x=1$
$\therefore \int_1^3 \frac{ c }{x} \cdot d x=1$
$\therefore c \int_1^3 \frac{1}{x} \cdot d x=1$
$\therefore c [\log x]_1^3=1$
$\therefore c [\log 3-\log 1]=1$
$\therefore c [\log 3-0]=1$
$\therefore c =\frac{1}{\log 3} $
$b. E ( X )=\int_{-\infty}^{\infty} x f(x)$
$ =\int_1^3 x f(x) \cdot d x$
$=\int_1^3 x \frac{ c }{x} \cdot d x$
$= c \int_1^3 1 \cdot d x$
$=\frac{1}{\log 3}[x]_1^3$
$=\frac{1}{\log 3}[3-1]$
$=\frac{2}{\log 3} $
$c. E \left( X ^2\right)=\int_{-\infty}^{\infty} x^2 f(x)$
$=\int_1^3 x^2 f(x) \cdot d x$
$=-\int_1^3 x^2 \cdot \frac{ c }{x} \cdot d x$
$= c \int_1^3 x \cdot d x$
$=\frac{1}{2 \log 3}\left[x^2\right]_1^3$
$=\frac{1}{2 \log 3}[9-1]$
$=\frac{8}{2 \log 3}$
$=\frac{4}{\log 3}$
$\therefore \operatorname{Var}( X )= E \left( X ^2\right)-[ E ( x )]^2$
$=\frac{4}{\log 3}-\left(\frac{2}{\log 3}\right)^2$
$=\frac{4}{(\log 3)}-\frac{4}{(\log 3)^2}$
$=\frac{4 \log 3-4}{(\log 3)^2}$
$=\frac{4(\log 3-1)}{(\log 3)^2}$
$F( x )=\int_1^x f(x) \cdot d x$
$=\int_1^x \frac{ c }{x} \cdot d x$
$= c \int_1^x \frac{1}{x} \cdot d x$
$= c [\log x]_1^x$
$= c [\log x -\log 1]$
$= c \log x $
$=\frac{\log x}{\log 3}$

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