A-Level CAIE Mathematics A2 6.2 Linear combinations of random variables Question Bank

A bus station has exactly four entrances. In the morning the numbers of passengers arriving at these entrances during a 10 -second period have the independent distributions $\operatorname{Po}(0.4), \operatorname{Po}(0.1), \operatorname{Po}(0.2)$ and Po(0.5).

Find the probability that the total number of passengers arriving at the four entrances to the bus station during a randomly chosen 1 -minute period in the morning is more than 3.

The random variable X has the distribution $\mathrm{N}\left(31.2,10.4^{2}\right)$. Two independent random values of X, denoted by $X_{1}$ and $X_{2}$, are chosen.

Find $\mathrm{P}\left(X_{1}>3 X_{2}\right)$.

The masses in kilograms of large and small bags of cement have the independent distributions N(50,2.4) and N(26,1.8) respectively.

Find the probability that the total mass of 5 randomly chosen large bags of cement is greater than the total mass of 10 randomly chosen small bags of cement.

Each year a transport firm uses X litres of gasoline and Y litres of diesel fuel, where X and Y have the independent distributions $X \sim \mathrm{~N}\left(10700,950^{2}\right)$ and $Y \sim \mathrm{~N}\left(13400,1210^{2}\right)$.

A teacher models the numbers of girls and boys who arrive late for her class on any day by the independent random variables $G \sim \operatorname{Po}(0.10)$ and $B \sim \operatorname{Po}(0.15)$ respectively.

The independent random variables X and Y have the distributions $\operatorname{Po}(1.9)$ and $\operatorname{Po}(2.2)$ respectively．