Study guides › 9231 FPS › Probability generating functions Maths4U Original · Learn, check, explain
The probability generating function — 9231 FPS
Read probabilities from coefficients and recover moments using derivatives.
1. Understand 2. See examples 3. Check answers 4. Explain your working Notes and key methods For a nonnegative integer-valued X, \(G_X(t)=E(t^X)=\sum p_xt^x\). The coefficient of \(\displaystyle t^{r}\) is \(\displaystyle P(X=r)\), and \(\displaystyle G(1)=1\).
Use \(E(X)=G'(1)\) and \(\operatorname{Var}(X)=G''(1)+G'(1)-[G'(1)]^2\). The second derivative gives E(X(X−1)), not \(\displaystyle E(X^{2})\), which explains the extra first-moment term.
Check the conditions X takes nonnegative integer values for an ordinary PGF. Moments require the relevant derivatives at \(\displaystyle t=1\) to exist. Three worked examples Example 1 · Probability generating functions X has PGF \(\displaystyle G_{X}(t)= \frac{5+ 6t+ 8t^{2}}{19}\).
Find \(\displaystyle P(X=2)\). Find E(X). Find Var(X). Find E(X(X−1)). Show the reasoning (a) \(\displaystyle [t^2]G_X(t)=\frac{8}{19}\)
Answer: \(\displaystyle \frac{8}{19}\).
(b) \(\displaystyle G_X'(1)=\frac{22}{19}\)
Answer: \(\displaystyle \frac{22}{19}\).
(c) \(\displaystyle G_X''(1)+G_X'(1)-[G_X'(1)]^2=\frac{238}{361}\)
Answer: \(\displaystyle \frac{238}{361}\).
(d) \(\displaystyle G_X''(1)=\frac{16}{19}\)
Answer: \(\displaystyle \frac{16}{19}\).
Example 2 · Probability generating functions X has PGF \(\displaystyle G_{X}(t)= \frac{6+ 7t+ 9t^{2}}{22}\).
Find \(\displaystyle P(X=2)\). Find E(X). Find Var(X). Find E(X(X−1)). Show the reasoning (a) \(\displaystyle [t^2]G_X(t)=\frac{9}{22}\)
Answer: \(\displaystyle \frac{9}{22}\).
(b) \(\displaystyle G_X'(1)=\frac{25}{22}\)
Answer: \(\displaystyle \frac{25}{22}\).
(c) \(\displaystyle G_X''(1)+G_X'(1)-[G_X'(1)]^2=\frac{321}{484}\)
Answer: \(\displaystyle \frac{321}{484}\).
(d) \(\displaystyle G_X''(1)=\frac{9}{11}\)
Answer: \(\displaystyle \frac{9}{11}\).
Example 3 · Probability generating functions X has PGF \(\displaystyle G_{X}(t)= \frac{7+ 8t+ 10t^{2}}{25}\).
Find \(\displaystyle P(X=2)\). Find E(X). Find Var(X). Find E(X(X−1)). Show the reasoning (a) \(\displaystyle [t^2]G_X(t)=\frac{2}{5}\)
Answer: \(\displaystyle \frac{2}{5}\).
(b) \(\displaystyle G_X'(1)=\frac{28}{25}\)
Answer: \(\displaystyle \frac{28}{25}\).
(c) \(\displaystyle G_X''(1)+G_X'(1)-[G_X'(1)]^2=\frac{416}{625}\)
Answer: \(\displaystyle \frac{416}{625}\).
(d) \(\displaystyle G_X''(1)=\frac{4}{5}\)
Answer: \(\displaystyle \frac{4}{5}\).
Common mistakes Do not mistake G″(1) for the second raw moment. Probabilities are coefficients, not PGF values at the outcome. Practise with answer checking Original exam-style problems Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.
Next skill: Mean (E(X)) and variance (Var(X)) using the probability generating function →
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