The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.6 )\).
(a) Find \(\mathrm { P } ( X \geq 8 )\).
(b) Find the value of \(\mathrm { E } ( X )\).
(c) Find the value of \(\operatorname { Var } ( X )\).
The random variable \(Y\) has the distribution \(\operatorname { Geo } ( p )\). It is given that \(\mathrm { P } ( Y < 4 ) = 0.986\) correct to 3 significant figures. Use an algebraic method to find the value of \(p\).
Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
State these two assumptions.
For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context.
Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\operatorname { Po } ( 0.8 )\).
(a) Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house.
(b) Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house.
Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house.
The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition.