A discrete random variable \(X\) has a probability function defined by
$$\mathrm { P } ( X = x ) = \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { x } } { x ! } \quad \text { for } x = 0,1,2,3,4 , \ldots \ldots$$
State the name of the distribution of \(X\).
Write down, in terms of \(\lambda\), expressions for \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).
Write down an expression for \(\mathrm { P } ( X = x + 1 )\), and hence show that
$$\mathrm { P } ( X = x + 1 ) = \frac { \lambda } { x + 1 } \mathrm { P } ( X = x )$$
The number of cars and the number of coaches passing a certain road junction may be modelled by independent Poisson distributions.
On a winter morning, an average of 500 cars per hour and an average of 10 coaches per hour pass this junction.
Determine the probability that a total of at least 10 such vehicles pass this junction during a particular 1 -minute interval on a winter morning.
On a summer morning, an average of 836 cars per hour and an average of 22 coaches per hour pass this junction.
Calculate the probability that a total of at most 3 such vehicles pass this junction during a particular 1 -minute interval on a summer morning. Give your answer to two significant figures.
(3 marks)