Geometric & Negative Binomial Distributions (Edexcel A-Level Further Mathematics): Quizzes

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The Negative Binomial Distribution
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What does the negative binomial distribution model?

Trials needed for fixed successes

In P(Y=x)=(x+r−1r−1)pr(1−p)xP(Y = x) = \binom{x + r - 1}{r - 1} p^r (1-p)^x, what does xx represent?

Number of failures

In the negative binomial PMF, what does rr represent?

Fixed number of successes

For Y∼NegBin(r,p)Y \sim \text{NegBin}(r, p), what is the mean?

r(1−p)p\frac{r(1-p)}{p}

For Y∼NegBin(r,p)Y \sim \text{NegBin}(r, p), what is the variance?

r(1−p)p2\frac{r(1-p)}{p^2}

If Y∼NegBin(r,p)Y \sim \text{NegBin}(r, p) and X∼Bin(n,p)X \sim \text{Bin}(n, p), how are they related?

P(Y≤n)=P(X≥r)P(Y \leq n) = P(X \geq r)

With p=0.6p=0.6, what is PP(exactly 4 shots for 2 successful shots)?

0.1728

How does negative binomial differ from binomial distribution?

It counts failures before successes

For r=5r=5 and p=0.3p=0.3, what is the mean number of failures?

11.67

For r=5r=5 and p=0.3p=0.3, what is the variance?

38.89

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Geometric & Negative Binomial Distributions

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