CAIE A-Level Further Math A2 4.5.3 Sums of Random Variables Questions

Practise using probability generating functions to combine independent random variables and derive distributions with Further Mathematics AS questions and mark schemes.

Exam points

Use probability generating functions to combine independent random variables.

[Maximum number: 3]

Keira has two unbiased coins. She tosses both coins. The number of heads obtained by Keira is denoted by X.

Find the probability generating function of Z, expressing your answer as a polynomial.

$\mathrm{G}_{Z}(t)=\left(\frac{1}{4}+\frac{1}{2} t+\frac{1}{4} t^{2}\right)\left(\frac{12}{36}+\frac{16}{36} t+\frac{7}{36} t^{2}+\frac{1}{36} t^{3}\right)$
Attempt to multiply their two PGF

$=\frac{1}{144}\left(12+40 t+51 t^{2}+31 t^{3}+9 t^{4}+t^{5}\right)$
Obtain quintic expression and collect terms

3

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CAIE A-Level Further Math A2 4.5.3 Sums of Random Variables Questions question 1