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A random variable $x$ has a probability density function $f(x) = \left\{ \begin{array}{1 1} k & \quad 0 < x < 2n \\ 0 & \quad \text{else where} \end{array} \right.$ Find $k$

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Toolbox:
  • The probability density function (continuous probability function $f(x)$ satisfies the following properties :
  • (i) $P(a\leq x\leq b)=\int_a^b f(x) dx$
  • (ii) $f(x)$ is non-negative for all real $x$
  • (iii) $\int_{-\infty}^\infty f(x) dx=1$
  • Also $P(x=a)=0$
  • $P(a\leq x\leq b)=P(a\leq x\leq b)$=P(a < x < b)
$\int_{-\infty}^{\infty} f(x) dx=1$
$\int_0^{2\pi} kdx=1$
Since $f(x)=0$ elsewhere
$\therefore kx\bigg]_0^{2\pi}=1$
$2\pi k=1$
$k=\large\frac{1}{2\pi}$
answered Sep 17, 2013 by sreemathi.v
 

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