Probability Generating Functions (PGFs) (Edexcel A-Level Further Mathematics): Quizzes

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Probability Generating Functions (PGFs)
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What is a PGF primarily used for?

Representing probabilities compactly

What is the definition of the PGF GX(t)G_X(t) for a discrete random variable XX?

GX(t)=∑x=0∞pxtxG_X(t) = \sum_{x=0}^\infty p_x t^x

What does tt represent in the PGF GX(t)G_X(t)?

A dummy variable

What is the value of GX(1)G_X(1) for any PGF?

11

What is the PGF for X∼B(n,p)X \sim B(n, p)?

[pt+(1−p)]n[pt + (1-p)]^n

What is the PGF for X∼Po(λ)X \sim \text{Po}(\lambda)?

eλ(t−1)e^{\lambda(t-1)}

What is the PGF for X∼Geom(p)X \sim \text{Geom}(p)?

p1−(1−p)t\frac{p}{1 - (1-p)t}

What is the PGF for X∼NegBin(r,p)X \sim \text{NegBin}(r, p)?

(p1−(1−p)t)r\left(\frac{p}{1 - (1-p)t}\right)^r

How do you find P(X=x)P(X = x) from a PGF?

1x!dxdtxGX(t)∣t=0\frac{1}{x!} \frac{d^x}{dt^x} G_X(t)|_{t=0}

For X∼Po(3)X \sim \text{Po}(3), what is GX′′(0)G_X''(0)?

9e−39e^{-3}

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