The random variable \(X\) has the binomial distribution \(\mathrm { B } ( 12,0 \cdot 3 )\). The independent random variable \(Y\) has the Poisson distribution \(\mathrm { Po } ( 4 )\). Find
\(\mathrm { E } ( X Y )\),
\(\quad \operatorname { Var } ( X Y )\).
The length of time a battery works, in tens of hours, is modelled by a random variable \(X\) with cumulative distribution function
$$F ( x ) = \begin{cases} 0 & \text { for } x < 0 \frac { x ^ { 3 } } { 432 } ( 8 - x ) & \text { for } 0 \leqslant x \leqslant 6 1 & \text { for } x > 6 \end{cases}$$
Find \(P ( X > 5 )\).
A head torch uses three of these batteries. All three batteries must work for the torch to operate. Find the probability that the head torch will operate for more than 50 hours.
Show that the upper quartile of the distribution lies between \(4 \cdot 5\) and \(4 \cdot 6\).
Find \(f ( x )\), the probability density function for \(X\).
Find the mean lifetime of the batteries in hours.
The graph of \(f ( x )\) is given below.
\includegraphics[max width=\textwidth, alt={}, center]{3cf25b54-d4a1-4d30-b632-3f6d3182a930-2_695_1463_1896_299}