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Choose ONE of the random variables from the options provided in each part. (a) Confirm the...

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Basic Math

Choose ONE of the random variables from the options provided ineach part. (a) Confirm the essential properties of the probabilityfunction for the binomial or Poisson or geometric random variable.[5 marks] (b) Derive the mean of the binomial or Poisson orgeometric random variable from first principles (i.e. using theprobability function and the definition of expectation). [7 marks](c) Confirm the essential properties of the probability densityfunction for the uniform or exponential random variable. [5 marks](d) Derive the cumulative distribution function for the uniform orexponential random variable. Show that this function meets thenecessary requirements for such a function (state what these are,and show that they are met). [8 marks] (e) Derive P(x1 < X

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