let x be a continuous random variable and suppose it has probability d

Question

Let X be a continuous random variable and suppose it has probability density function of the form f(x)=\left\{\begin{array}{ll}
\alpha(1-x)^{n-1} & \text { for } 0 \leq x \leq 1 \\
0 & \text { otherwise }
\end{array}\right. \text { where } n \text { is a known integer and } n>0 \text {. } \text { 1. Find the value for } \alpha \text { so that } f(x) \text { is in fact a probability density function. } Derive the corresponding cumulative distribution function, F(x). 3. Find a mathematical expression for the 0.25 quantile of X. That is find m sothat F(m) = 0.25.(5 marks) . Assuming that n = 5, calculate P(0.5 < X < 0.75).