Answer to Question #197777 in Statistics and Probability for Ranjith

Question

(b) (i) A random variable 'X' has the density function:

f(x) = K. — in- < x <

Answer 1

(b) Probability density function of the random variable $X$ is given by:

$f(x)=Kx(2-x)$ , $\ 0<x<2$

We know that for $p.d.f$ ,

$\int _{-\infty}^{\infty} f(x)dx=1$

$\Rightarrow \int _ {0}^{2} Kx(2-x)dx=1$

$\Rightarrow K\cdot\int _ {0}^{2}(2x-x^2)dx=1$

$\Rightarrow K\cdot |2.\frac{x^2}{2}-\frac {x^3}{3}|_{0}^{2}=1$

$\Rightarrow K\cdot[4-\frac{8}{3}]=1$

$\Rightarrow K=\dfrac{3}{4}\\$

Hence the value of K is $\boxed{K=\dfrac{3}{4}}$