Probability Distributions — Book-Back Answer Key (Exercise 11.6)
Probability Distributions — Book-Back Answer Key (Exercise 11.6)
All twenty multiple-choice answers below were re-worked independently. Each one agreed with the official key, so no corrections were required.
| Q | Answer | Why |
|---|---|---|
| 1 | (2) mean exists, variance does not | \(E(X)=\int_1^\infty \tfrac{2}{x^2}\,dx=2\) is finite, but \(E(X^2)=\int_1^\infty \tfrac{2}{x}\,dx\) diverges. |
| 2 | (4) \(\tfrac{l}{2},\ \tfrac{l^2}{12}\) | Uniform on \((0,l)\): \(\mu=\tfrac{l}{2}\), \(E(X^2)=\tfrac{l^2}{3}\), so \(\sigma^2=\tfrac{l^2}{3}-\tfrac{l^2}{4}=\tfrac{l^2}{12}\). |
| 3 | (2) \(-\tfrac{19}{6}\) | \(E=\tfrac{1}{6}(36)-\tfrac{1}{6}(1+4+9+16+25)=6-\tfrac{55}{6}=-\tfrac{19}{6}\). |
| 4 | (4) \(4\) | Sum \(=7\) from \((3,4),(4,3),(5,2),(6,1)\) — four pairs. |
| 5 | (4) \(2\) | \(\sigma=\sqrt{npq}=\sqrt{25(0.8)(0.2)}=\sqrt{4}=2\). |
| 6 | (2) \(2i-n\) | With \(i\) heads, the difference is \(i-(n-i)=2i-n\) for \(i=0,\dots,n\). |
| 7 | (4) \(16\) and \(24\) | A uniform density needs \(b-a=12\); here \(24-16=8\neq 12\). |
| 8 | (3) \(40.75,\ 40\) | Size-biased \(E(X)=\tfrac{42^2+36^2+34^2+48^2}{160}=40.75\); uniform \(E(Y)=\tfrac{160}{4}=40\). |
| 9 | (2) \(\tfrac{2}{13}\) | With replacement, \(E(X)=2\cdot\tfrac{1}{13}=\tfrac{2}{13}\) by linearity. |
| 10 | (1) \(\tfrac{11}{243}\) | \(B(5,\tfrac13)\): \(P(X\ge4)=\tfrac{10}{243}+\tfrac{1}{243}=\tfrac{11}{243}\). |
| 11 | (4) \(\tfrac{1}{81}\) | \(E=3\operatorname{Var}\Rightarrow q=\tfrac13\); with \(n=4\), \(P(X=0)=\left(\tfrac13\right)^4=\tfrac{1}{81}\). |
| 12 | (4) \(\binom{10}{5}\left(\tfrac35\right)^5\left(\tfrac25\right)^5\) | \(np=6,\ npq=2.4\Rightarrow q=\tfrac25,\ p=\tfrac35,\ n=10\). |
| 13 | (1) \(1\) and \(-1\) | \(F(\infty)=a=1\), \(F(0)=a+b=0\Rightarrow b=-1\), giving \(1-e^{-x}\). |
| 14 | (2) \(2\) | \(q=\tfrac{\operatorname{Var}}{\text{mean}}=\tfrac12\); \(np=8\!\left(\tfrac12\right)=4=2k\Rightarrow k=2\). |
| 15 | (1) I and III | Bulb lifetime and call duration are measured (continuous); a count is discrete. |
| 16 | (1) \(1\) | \(\sum_{x\ge1} a\left(\tfrac12\right)^x=a\cdot 1=1\Rightarrow a=1\). |
| 17 | (4) \(\tfrac{2}{3}\) | \(15k=1\Rightarrow k=\tfrac{1}{15}\); \(E(X)=10k=\tfrac{10}{15}=\tfrac23\). |
| 18 | (4) \(0.96\) | \(\operatorname{Var}(X)=pq=0.24\); \(\operatorname{Var}(2X-3)=4(0.24)=0.96\). |
| 19 | (2) \(0.25\) | \(9p^2=q^2\Rightarrow 3p=1-p\Rightarrow p=\tfrac14\). |
| 20 | (1) \(\tfrac{57}{8000}\) | \(B(3,\tfrac1{20})\): \(P(X=2)=3\left(\tfrac1{20}\right)^2\left(\tfrac{19}{20}\right)=\tfrac{57}{8000}\). |