CAIE A-Level Further Mathematics 4.5.3 PGFs for Sums of Random Variables

Practise multiplying probability generating functions for independent sums and using the result for probabilities and moments.

Exam points

multiply component PGFs and expand the product to obtain the sum's PGF
read P(Z=k) from the coefficient of t^k in the expanded generating function
differentiate at t=1 to relate the expectation of a sum to component means

[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.

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

M1

Attempt to multiply their two PGF

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

M1 A1

Obtain quintic expression and collect terms

3

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4.5.3—Sums of random variables question 1