Question

Choose ONE of the random variables from the options provided in each part. (a) Confirm the essential properties of the probability function for the binomial or Poisson or geometric random variable. [5 marks] (b) Derive the mean of the binomial or Poisson or geometric random variable from first principles (i.e. using the probability function and the definition of expectation). [7 marks] (c) Confirm the essential properties of the probability density function for the uniform or exponential random variable. [5 marks] (d) Derive the cumulative distribution function for the uniform or exponential random variable. Show that this function meets the necessary requirements for such a function (state what these are, and show that they are met). [8 marks] (e) Derive P(x1 < X < x2) for the uniform or exponential random variable. Your answer should be a function of x1, x2 and the parameters of the distribution you choose. You may use your result in (d), but if you choose a different random variable, you must start from f(x). [5 marks] (f) Derive the mean of the uniform or exponential or normal random variable by any method. [5 marks] (g) Derive the moment generating function for any one of the random variables listed in the test resource. [6 marks]Derive E(X2 ) for any one of the random variables listed in the test resource, from first principles (i.e. using f(x) and the definition of expectation) or by using its moment generating function. Hence or otherwise, derive the variance of that random variable. You may assume linearity of expectation as it app

Answer 1