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Question 3: Continuous Random Variable (20 marks) Let X be a continuous random variable and suppose it has probability density function of the form for 0 0. 1. Find the value for a so that f (x) is in fact a probability density function. (5 marks) =f (x) / n( 1-x) "-'dx = - [0- 1] = - (-1) =1 The probability density function (pdf) f(x) of a continuous random variable X is defined as the derivative of the cdf F(x): f(x)=ddxF(x). The pdf f(x) has two important properties: f(x)20, for all x. 2. Derive the corresponding cumulative distribution function, F (x). (5 marks) f(x) = P(xSx)= / m(1-ne-Id [osxsi] = 1 _(1-1)" "= [(1- x)" - 1] f(x) = 1- (1-1)*3. Find a mathematical expression for the 0.25 quantile of X. That is find m so that F (m) = 0.25. (5 marks) F( m) = 0.50 =1- (1-m) " = 0.5 = 1- (0.5) " = m 4. Assuming that n = 5, calculate P (0.5