Please solve the question below with precision

(i) Derive the median of a Pareto distribution with porometers o and 2. [3] Lel u = 2 and 2. = 3. (ii) Comment on the skewness of this Pareto distribution. [3] [Total 6] A portfolio of insurance policies has two types of claims: Loss amounts for Type I claims are exponentially distributed with mcan 120. Loss amounts for Type II claims are exponentially distributed with mean 1 10. 25% of claims are Type I, and 75% are Type II. (i) Calculate the mean and variance of the loss amount for a randomly chosen claim. (3] An actuary wants to model randomly chosen claims using an exponential distribution as an approx imation. (ii) Explain whether this is a good approximation. [2] [Total 5] A child playing a game believes that a six sided die is unfair, and that he has a probability p > 1/6 of predicting the outcome of any given throw. His mother is less sure, and her prior beliefs about p are as follows: a 1/3 chance that p = 2/6 and . a 2/3 chance that p = 1/6 The child accurately predicts the results of 4 out of 10 dice throws. Calculate the posterior probability that p = 1/6. [6]