2 MARKS are reserved for overall readability, structure and functionality of the uploaded files. In total 25 marks. 1. Consider a random variable X with the probability distribution fx (2) = 7 ( + B) r +1 where r and B are some suitable positive real parameters, and x 2 0. (a) From the normalization condition, determine c such that fx is a proper probability distribution (PDF). 2 MARKS (b) Calculate the mean and variance. 2+2 MARKS (c) Now we may need to take care whether restrictions are to be made on the parameters r and B: Do mean and variance assume finite values for all positive values of r and all positive values of B? Justify your answers. 2 MARKS (d) Calculate the numerical value of mean and variance (when these are finite, check!) for r := F and = 2 + 0.ABCDEF where ABCDEF are the digits of your 6-digit candidate number. 2 MARKS 2. Consider a mixed random variable X with an event space of a single discrete event at x = -(B + 1) and the piecewise constant continuous PDF ff (2 ) = 0 , x 2+ A . The mixed probability density function then follows (here () dentotes the Dirac delta function) fx ( ) = c(8 (x + B + 1) + rfx(x) ), where r is a positive real parameter, and A and B are digits of your 6-digit candidate number ABCDEF. (a) Calculate c such that fx (x) is normalized. 2 MARKS (b) Calculate the mean / = E[X] (which will depend on the parameter r). 3 MARKS (c) Determine the value of r for which & = 0. 1 MARK (d) Calculate the variance o', for this specific value of r (or in the case of a general r if you prefer). (it is acceptable if you insert r and c from above, without simplifying further. But you need to take care of all integrations that occur.) 2 MARKS (e) Start calculating the excess kurtosis, using the definition, and performing all integrations, for the specific value of r found above. 2 MARKS (f) Find the cumulative distribution function (CDF). It is sufficient to draw it graphically. 2 MARK (g) Find the median, for the parameters above. 1 MAR