### Mathematical Statistics A – Example Sheet 1

Thought I may as well put answers to the A & C sections here, for simple-ish reflection. And to notice when I can’t recall how to do things, at all.

QA1: We compute the cumulative distribution function by integrating the density function:We argue Y is discrete because…it takes a finite number of possible values? It’s a step function? I’m not sure. Its support is {0,2} (I hope), pmf and cdf are:

QA2 is simple infinite sum fiddling combined with the identity for .

QC5:

By the inverse function theorem (differentiation). Y is continuous as the functions that make up Y are continuous – additionally, as g is an increasing bijection it is continuous and differentiable, and its derivative is always positive, so the pdf of Y is not undefined (g’ always nonzero).

For the last part, simply plug in g(X) = aX + b to what you’ve already worked out, and find Y is the Gaussian (given X is the standard Gaussian).

QC6:

Find the mgf of the exponential in the normal way; use the same trick as Q4 to find the mgf of . Find the mgf of using the pdf; as they are equal use uniqueness to conclude the distribution is correct.

Integrate from t to infinity; receive sum from 0 to n. Observe that as the number of photons detected must be an integer, . Plug in your previously calculated values for the answer.

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