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Further Probability and Statistics

Quick questions on Special discrete distributions explained: H2 Further Mathematics

9short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the geometric distribution?
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If XX is the number of trials up to and including the first success, then X∼Geometric(p)X \sim \text{Geometric}(p) with
What is the negative binomial distribution?
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If YY is the number of trials up to and including the rrth success, then YY is negative binomial with parameters rr and pp:
What are the relationships between the distributions?
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The geometric distribution is the negative binomial with r=1r = 1. And the negative binomial counting trials to the rrth success is the sum of rr independent geometric variables, which is why its mean rp\tfrac{r}{p} and variance r(1−p)p2\tfrac{r(1-p)}{p^2} are rr times the geometric values.
What is wrong exponent in the geometric formula?
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It is (1−p)x−1p(1 - p)^{x-1}p: x−1x - 1 failures then a success, not (1−p)xp(1-p)^x p.
What is wrong combination in the negative binomial?
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Use (y−1r−1)\binom{y - 1}{r - 1}, since the rrth success is fixed on the last trial and only the first y−1y - 1 trials are arranged.
What is mean of geometric as pp?
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The mean is 1p\tfrac{1}{p}, not pp; a small success probability means many trials on average.
What is q1?
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Write the geometric probability P(X=x)\mathrm{P}(X = x) for success probability pp. [1 mark]
What is q2?
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State the mean of a geometric distribution with parameter pp. [1 mark]
What is q3?
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What distribution counts the number of trials to the rrth success? [1 mark]

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