Geometric and negative binomial distributions questions

Questions to attempt

6 questions on Geometric and negative binomial distributions, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. FS1-3.1 · Geometric and negative binomial distributions.

   Q1

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   1.

   Independent trials have success probability $0.4$. Find the probability that the third success occurs on trial $5$, and find the probability that the third success occurs after trial $6$.

   (4)

   (Total for Question 1 is 4 marks)

   Reveal answer and method

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2. FS1-3.1 · Geometric and negative binomial distributions.

   Q2

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   2.

   Independent trials have success probability $0.3$. Find the probability that the first success occurs on trial $4$.

   (2)

   (Total for Question 2 is 2 marks)

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3. FS1-3.2 · Mean and variance of a geometric distribution with parameter p.

   Q3

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   3.

   A geometric random variable has mean $8$. Find $p$, its variance and its standard deviation.

   (4)

   (Total for Question 3 is 4 marks)

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4. FS1-3.2 · Mean and variance of a geometric distribution with parameter p.

   Q4

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   4.

   The random variable $X$ is geometric with parameter $p=0.2$ and counts the trial of the first success. Find its mean and variance.

   (2)

   (Total for Question 4 is 2 marks)

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5. FS1-3.3 · Mean and variance of negative binomial distribution.

   Q5

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   5.

   A negative binomial random variable has mean $15$ and variance $30$. Find $p$ and $r$, then calculate $P(X=7)$.

   (6)

   (Total for Question 5 is 6 marks)

   Reveal answer and method

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6. FS1-3.3 · Mean and variance of negative binomial distribution.

   Q6

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   6.

   The random variable $X$ counts the trial on which the fourth success occurs, with success probability $0.25$. Find $E(X)$ and $\operatorname{Var}(X)$.

   (2)

   (Total for Question 6 is 2 marks)

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FS1-3 Geometric and negative binomial distributions revision notes