Probability generating functions questions

Questions to attempt

6 questions on Probability generating functions, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. FS1-7.1 · Definitions, derivations and applications. Use of the probability generating function for the negative binomial, geometric, binomial and Poisson 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.

   A geometric random variable $X$ has parameter $p=0.4$ and counts the trial of the first success. Show that the probability generating function of $X$ is $\dfrac{0.4t}{1-0.6t}$ and hence find $P(X=5)$.

   (5)

   (Total for Question 1 is 5 marks)

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2. FS1-7.1 · Definitions, derivations and applications. Use of the probability generating function for the negative binomial, geometric, binomial and Poisson 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.

   Given $X\sim\operatorname{Bin}(4,0.3)$, write down $G_X(t)$ and use its coefficient of $t^2$ to find $P(X=2)$.

   (3)

   (Total for Question 2 is 3 marks)

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3. FS1-7.2 · Use to find the mean and variance.

   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 random variable has PGF $G_X(t)=\dfrac{2t}{3-t}$. Use the PGF to find its mean and variance.

   (5)

   (Total for Question 3 is 5 marks)

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4. FS1-7.2 · Use to find the mean and variance.

   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 PGF of $X$ is $G_X(t)=(0.8+0.2t)^6$. Use derivatives of the PGF to find $E(X)$ and $\operatorname{Var}(X)$.

   (4)

   (Total for Question 4 is 4 marks)

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5. FS1-7.3 · Probability generating function of the sum of independent random variables.

   Q5

   ORIGINAL

   Year 2

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

   5.

   Independent variables satisfy $X\sim\operatorname{Bin}(5,0.4)$ and $Y\sim\operatorname{Bin}(7,0.4)$. Use PGFs to identify $S=X+Y$ and calculate $P(S=3)$.

   (5)

   (Total for Question 5 is 5 marks)

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6. FS1-7.3 · Probability generating function of the sum of independent random variables.

   Q6

   ORIGINAL

   Year 2

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

   6.

   Independent variables have distributions $X\sim\operatorname{Po}(1.4)$ and $Y\sim\operatorname{Po}(2.6)$. Use PGFs to identify the distribution of $X+Y$ and find $P(X+Y=0)$.

   (4)

   (Total for Question 6 is 4 marks)

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FS1-7 Probability generating functions revision notes