Suppose that [math]X[/math] is a random variable which represents the number of calls coming in to a police station in a one-minute interval. In the text, we showed that [math]X[/math] could be modelled using a Poisson distribution with parameter [math]\lambda[/math], where this parameter represents the average number of incoming calls per minute. Now suppose that [math]Y[/math] is a random variable which represents the number of incoming calls in an interval of length [math]t[/math]. Show that the distribution of [math]Y[/math] is given by
i.e., [math]Y[/math] is Poisson with parameter [math]\lambda t[/math]. Hint: Suppose a Martian were to observe the police station. Let us also assume that the basic time interval used on Mars is exactly [math]t[/math] Earth minutes. Finally, we will assume that the Martian understands the derivation of the Poisson distribution in the text. What would she write down for the distribution of [math]Y[/math]?
The Poisson distribution with parameter [math]\lambda = .3[/math] has been assigned for the outcome of an experiment. Let [math]X[/math] be the outcome function. Find [math]P(X = 0)[/math], [math]P(X = 1)[/math], and [math]P(X \gt 1)[/math].
On the average, only 1 person in 1000 has a particular rare blood type.
- Find the probability that, in a city of 10,00 people, no one has this blood type.
- How many people would have to be tested to give a probability greater than 1/2 of finding at least one person with this blood type?
Write a program for the user to input [math]n[/math], [math]p[/math], [math]j[/math] and have the program print out the exact value of [math]b(n, p, k)[/math] and the Poisson approximation to this value.
Assume that, during each second, a Dartmouth switchboard receives one call with probability .01 and no calls with probability .99. Use the Poisson approximation to estimate the probability that the operator will miss at most one call if she takes a 5-minute coffee break.
The probability of a royal flush in a poker hand is [math]p =1/64940[/math]. How large must [math]n[/math] be to render the probability of having no royal flush in [math]n[/math] hands smaller than [math]1/e[/math]?
A baker blends 600 raisins and 400 chocolate chips into a dough mix and, from this, makes 500 cookies.
- Find the probability that a randomly picked cookie will have no raisins.
- Find the probability that a randomly picked cookie will have exactly two chocolate chips.
- Find the probability that a randomly chosen cookie will have at least two bits (raisins or chips) in it.
The probability that, in a bridge deal, one of the four hands has all hearts is approximately [math]6.3 \times 10^{-12}[/math]. In a city with about 50,000 bridge players the resident probability expert is called on the average once a year (usually late at night) and told that the caller has just been dealt a hand of all hearts. Should she suspect that some of these callers are the victims of practical jokes?
An advertiser drops 10,00 leaflets on a city which has 2000 blocks. Assume that each leaflet has an equal chance of landing on each block. What is the probability that a particular block will receive no leaflets?