Discrete distribution with probabilities in a constant ratio: unknown probabilities, the constant and the expected value
Discrete distribution with probabilities in a constant ratio: unknown probabilities, the constant and the expected value
6 marks
A discrete random variable \(X\) takes the values \(1,2,3,4\) and \(5\). Its probabilities satisfy
\(\mathrm{P}(X=1)=k\quad\text{and}\quad\mathrm{P}(X=r+1)=\dfrac13\,\mathrm{P}(X=r)\ \text{ for}\hspace{0.5em} r=1,2,3,4,\)
where \(k\) is a constant. The probability distribution of \(X\) is shown in the table.
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
|---|---|---|---|---|---|
| \(\mathrm{P}(X=x)\) | \(k\) | \(a\) | \(\frac{9}{121}\) | \(\frac{k}{27}\) | \(b\) |
- Write down, in terms of \(k\), an expression for
- \(a\);
- \(b\).[2]
- Find \(k\).[2]
- Hence find \(\mathrm{E}(X)\).[2]
Solution:
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Answer:(a)(i) \(a=\dfrac{k}{3}\) (a)(ii) \(b=\dfrac{k}{81}\) (b) \(k=\dfrac{81}{121}\) (c) \(\mathrm{E}(X)=\dfrac{179}{121}\approx1.48\)
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