Expected value and variance of a derived variable
Expected value and variance of a derived variable
IB Year 6 | Grade 12
A fair coin and a fair six-faced die are thrown simultaneously. The random variable \(X\) is defined as follows:
If the coin shows a head, then \(x\) is the score on the die.
If the coin shows a tail, then \(x\) is twice the score on the dice.
- Find \(\mu\), the expected value of \(X\) and show that \(\mathrm{P}(X < \mu) = \frac{7}{12}\).
- Show that \(\mathrm{Var}(X) = \frac{497}{48}\).
Solution:
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Answer:(i) \(\mu=\frac{21}{4}\), \(P(X<\mu)=\frac7{12}\); (ii) \(\operatorname{Var}(X)=\frac{497}{48}\).
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