Question

Define the density function (f(x)) as below: f(x) = cx'for 0<x<2,0 otherwise Where c was determined above. What is the probability this random variable takes a value between 1 and 1.5?

Answer 1

$\\ For\ probability \ density\ function\\ \\ \int f(x)\ dx=1\\\\ \Rightarrow \int^2_0 cx^3\ dx=1\\\\ \Rightarrow c\int^2_0 x^3\ dx=1\\\\ \Rightarrow c\left [ \frac{x^4}{4} \right ]^2_0=1\\\\ \Rightarrow c(4)=1\\\\ \Rightarrow c=\frac{1}{4}$

$\\ Probability \ this\ random\ variable\ take\ a\ value\ between\ 1\ and\ 1.5\\\\ \int^{1.5}_1f(x)\ dx\\\\ =\frac{1}{4}\left [ \frac{x^4}{4} \right ]^{1.5}_1\\\\\ =\frac{1}{4}(1.265625-0.25)\\ \\ =0.2539$