1 In a certain country, any baby born is equally likely to be a boy or a girl, independently for all births. The birthweight of a baby boy is given by the continuous random variable $X _ { B }$ with probability density function (pdf) $\mathrm { f } _ { B } ( x )$ and cumulative distribution function (cdf) $\mathrm { F } _ { B } ( x )$. The birthweight of a baby girl is given by the continuous random variable $X _ { G }$ with pdf $\mathrm { f } _ { G } ( x )$ and cdf $\mathrm { F } _ { G } ( x )$.

The continuous random variable $X$ denotes the birthweight of a baby selected at random.\\
(i) By considering

$$\mathrm { P } ( X \leqslant x ) = \mathrm { P } ( X \leqslant x \mid \text { boy } ) \mathrm { P } ( \text { boy } ) + \mathrm { P } ( X \leqslant x \mid \text { girl } ) \mathrm { P } ( \text { girl } ) ,$$

find the cdf of $X$ in terms of $\mathrm { F } _ { B } ( x )$ and $\mathrm { F } _ { G } ( x )$, and deduce that the pdf of $X$ is

$$\mathrm { f } ( x ) = \frac { 1 } { 2 } \left\{ \mathrm { f } _ { B } ( x ) + \mathrm { f } _ { G } ( x ) \right\} .$$

(ii) The birthweights of baby boys and girls have means $\mu _ { B }$ and $\mu _ { G }$ respectively. Deduce that

$$\mathrm { E } ( X ) = \frac { 1 } { 2 } \left( \mu _ { B } + \mu _ { G } \right) .$$

(iii) The birthweights of baby boys and girls have common variance $\sigma ^ { 2 }$. Find an expression for $\mathrm { E } \left( X ^ { 2 } \right)$ in terms of $\mu _ { B } , \mu _ { G }$ and $\sigma ^ { 2 }$, and deduce that

$$\operatorname { Var } ( X ) = \sigma ^ { 2 } + \frac { 1 } { 4 } \left( \mu _ { B } - \mu _ { G } \right) ^ { 2 } .$$

(iv) A random sample of size $2 n$ is taken from all the babies born in a certain period. The mean birthweight of the babies in this sample is $\bar { X }$. Write down an approximation to the sampling distribution of $\bar { X }$ if $n$ is large.\\
(v) Suppose instead that a stratified sample of size $2 n$ is taken by selecting $n$ baby boys at random and, independently, $n$ baby girls at random. The mean birthweight of the $2 n$ babies in this sample is $\bar { X } _ { s t }$. Write down the expected value of $\bar { X } _ { s t }$ and find the variance of $\bar { X } _ { s t }$.\\
(vi) Deduce that both $\bar { X }$ and $\bar { X } _ { s t }$ are unbiased estimators of the population mean birthweight. Find which is the more efficient.

\hfill \mbox{\textit{OCR MEI S4 2012 Q1 [24]}}