Extra Probability (HL)
IB Mathematics AA · HL · Shadow Worksheet
Practice
Question 1
If \(X\sim ext{Poisson}(3)\), find \(P(X=4)\), \(P(X\leq2)\), and \(E(X^2)\).
Poisson formula: \(P(X=k)=\dfrac{e^{-\lambda}\lambda^k}{k!}\), \(\lambda=3\)
\(P(X=4)\)
\(\dfrac{e^{-3}\cdot81}{24}=\dfrac{81e^{-3}}{24}pprox0.168\)
\(P(X\leq2)=P(0)+P(1)+P(2)\)
\(e^{-3}(1+3+ frac{9}{2})=e^{-3} imes8.5pprox0.423\)
\(E(X^2)= ext{Var}(X)+[E(X)]^2=3+9\)
Answer
\(P(X=4)pprox0.168\); \(P(X\leq2)pprox0.423\); \(E(X^2)=12\)
Question 2
Events A and B satisfy \(P(A)= frac{1}{3}\), \(P(B|A)= frac{1}{2}\), \(P(B|A')= frac{1}{4}\). Find \(P(A|B)\).
Total probability
\(P(B)= frac{1}{3} imes frac{1}{2}+ frac{2}{3} imes frac{1}{4}= frac{1}{6}+ frac{1}{6}= frac{1}{3}\)
Bayes
\(P(A|B)=\dfrac{P(B|A)P(A)}{P(B)}=\dfrac{ frac{1}{2} imes frac{1}{3}}{ frac{1}{3}}\)
Answer
\(P(A|B)=\dfrac{1}{2}\)
Question 3
A continuous random variable \(X\) has PDF \(f(x)=\lambda e^{-\lambda x}\) for \(x\geq0\) (Exponential distribution). Show that \(E(X)=\dfrac{1}{\lambda}\) and find the median.
\(E(X)=\displaystyle\int_0^\infty x\lambda e^{-\lambda x}\,dx\)
IBP with \(u=x,\;dv=\lambda e^{-\lambda x}\): \([-xe^{-\lambda x}]_0^\infty+\displaystyle\int_0^\infty e^{-\lambda x}\,dx=0+[- frac{1}{\lambda}e^{-\lambda x}]_0^\infty= frac{1}{\lambda}\) ✓
Median: solve \(\displaystyle\int_0^m\lambda e^{-\lambda x}\,dx= frac{1}{2}\)
\(1-e^{-\lambda m}= frac{1}{2}\Rightarrow m=\dfrac{\ln2}{\lambda}\)
Answer
Median \(=\dfrac{\ln2}{\lambda}\)
Question 4
Two independent random variables \(X\sim N(10,4)\) and \(Y\sim N(6,9)\). Find \(P(X>Y)\).
Let \(D=X-Y\)
\(D\sim N(10-6,\;4+9)=N(4,13)\)
\(P(D>0)\)
\(Z=\dfrac{0-4}{\sqrt{13}}pprox-1.109\Rightarrow P(Z>-1.109)=P(Z<1.109)pprox0.866\)
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