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Worksheet
Discrete Random Variables
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Name
Question 1
The discrete random variable \(X\) has the following distribution:
\[\begin{array}{c|ccccc}x&1&2&3&4&5\\hline P(X=x)&0.1&0.2&k&0.3&0.15\end{array}\]
(a)Find \(k\).
(b)Find \(E(X)\) and \(\text{Var}(X)\).
(c)Find \(E(3X-2)\) and \(\text{Var}(3X-2)\).
(a)
Probabilities sum to 1
\(k=1-0.75=0.25\)
Answer
\(k=0.25\)
(b)
\(E(X)\)
\(1(0.1)+2(0.2)+3(0.25)+4(0.3)+5(0.15)=0.1+0.4+0.75+1.2+0.75=3.2\)
\(E(X^2)\)
\(1(0.1)+4(0.2)+9(0.25)+16(0.3)+25(0.15)=0.1+0.8+2.25+4.8+3.75=11.7\)
\(\text{Var}(X)\)
\(11.7-3.2^2=11.7-10.24\)
Answer
\(E(X)=3.2\); \(\text{Var}(X)=1.46\)
(c)
Answer
\(E(3X-2)=3(3.2)-2=7.6\); \(\text{Var}(3X-2)=9\times1.46=13.14\)
Question 2
Two fair dice are rolled. \(X\) is the absolute difference of the scores. Find \(P(X=2)\) and \(E(X)\).
\(P(X=2)\)
Pairs where \(|a-b|=2\): (1,3),(2,4),(3,5),(4,6),(3,1),(4,2),(5,3),(6,4) — 8 outcomes. \(P=\tfrac{8}{36}=\tfrac{2}{9}\)
\(E(X)\)
Distribution: \(P(0)=\tfrac{6}{36},P(1)=\tfrac{10}{36},P(2)=\tfrac{8}{36},P(3)=\tfrac{6}{36},P(4)=\tfrac{4}{36},P(5)=\tfrac{2}{36}\)
\(E(X)=\tfrac{0+10+16+18+16+10}{36}\)
Answer
\(P(X=2)=\dfrac{2}{9}\); \(E(X)=\dfrac{70}{36}=\dfrac{35}{18}\approx1.94\)
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