2025 NYJC P2 Q7
2025 NYJC P2 Q7
9 marks
The random variable \(S\) is the number of successes in \(5\) independent trials of an experiment in which the probability of success in any trial is \(\frac{1}{3}\). The random variable \(D\) is the difference between the number of successes and the number of failures in \(5\) such trials.
- State the values that \(D\) can take.[1]
- Show that \(\mathrm{P}(D=1)=\frac{40}{81}\).[1]
- By finding the probability distribution of \(D\), find the exact value of \(\mathrm{E}(D^2)\).[3]
- Find the exact values of \(\mathrm{E}(S)\) and \(\mathrm{E}(S^2)\).[2]
- Hence, by showing that \(D^2 = 4S^2 - 20S + 25\), verify the correctness of the value of \(\mathrm{E}(D^2)\) found in part (c).[2]
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Answer:(a) \(D=1,3,5\) (b) \(\mathrm P(D=1)=\dfrac{40}{81}\) (c) \(\mathrm E(D^2)=\dfrac{65}{9}\) (d) \(\mathrm E(S)=\dfrac53\), \(\mathrm E(S^2)=\dfrac{35}{9}\) (e) \(D^2=4S^2-20S+25\), giving \(\mathrm E(D^2)=\dfrac{65}{9}\)
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