Probability Generating Functions
IB Mathematics AA · HL · Shadow Worksheet
Practice
Question 1
The PGF of a random variable \(X\) is \(G_X(t)=\dfrac{t^3}{8}+\dfrac{3t^2}{8}+\dfrac{3t}{8}+\dfrac{1}{8}\).
(a)Write down the probability distribution of \(X\).
(b)Find \(E(X)\) using \(G'_X(1)\).
(a)
Coefficient of \(t^k\) is \(P(X=k)\)
Answer
\(P(X=0)= frac{1}{8},\;P(X=1)= frac{3}{8},\;P(X=2)= frac{3}{8},\;P(X=3)= frac{1}{8}\)
(b)
Differentiate
\(G'(t)= frac{3t^2}{8}+ frac{6t}{8}+ frac{3}{8};\;G'(1)= frac{3+6+3}{8}\)
Answer
\(E(X)=\dfrac{3}{2}\)
Question 2
A random variable \(X\sim ext{Geometric}(p)\) has PGF \(G_X(t)=\dfrac{pt}{1-(1-p)t}\). Show that \(E(X)=\dfrac{1}{p}\).
Differentiate using quotient rule
\(G'(t)=\dfrac{p[1-(1-p)t]+pt(1-p)}{[1-(1-p)t]^2}=\dfrac{p}{[1-(1-p)t]^2}\)
Evaluate at \(t=1\)
\(G'(1)=\dfrac{p}{p^2}=\dfrac{1}{p}\) ✓
Question 3
If \(X\sim B(n,p)\) has PGF \(G_X(t)=(q+pt)^n\) where \(q=1-p\), find \( ext{Var}(X)\) using \(G''_X(1)\).
Second derivative
\(G''(t)=n(n-1)p^2(q+pt)^{n-2};\;G''(1)=n(n-1)p^2\)
Variance formula
\( ext{Var}(X)=G''(1)+G'(1)-[G'(1)]^2=n(n-1)p^2+np-n^2p^2=np-np^2=np(1-p)\)
Answer
\( ext{Var}(X)=npq\) ✓
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