Question

Suppose you are used to passing by, on average, 7 turtles on your one-mile walking route. You wish to come up with a pmf for the number of turtles Y that you will pass by on future walks on this route. Suppose you assume that with each 5-foot stride you will pass by a turtle with probability 7/1056 and that the event of your passing by a turtle in one stride is independent of the event of your passing by a turtle in any other stride. Recall: 1 mile = 5280 ft .

(a) Under these assumptions, what is the probability distribution of Y ?

(b) Under these assumptions, give EY

. (c) Under these assumptions, give P(Y ≤ 3).

(d) Now compute P(Y ≤ 3) treating Y as though it were a Poisson(λ) rv with λ = 7

(e) Explain what you think is the point of this question

Answer 1

(a)

We know, 1 mile = 5280 ft

We take 5 foot stride as an unit.

So, there are 5280/5 = 1056 units of 5 foot stride each.

We can model the given situation using Binomial distribution as follows.

Suppose, random number Y denotes number of strides in which turtle is passed by.

Passing by a turtle in one stride is independent of passing by a turtle in any other stride.

We define passing by a turtle as success.

In each unit of 5 foot stride, probability of success is 7/1056.

(b)

We know, for Binomially distributed random variable

(c)

Required probability is given by

   [Using R-code 'pbinom(3,1056,7/1056)']

(d)

Approximating to Poisson distribution we have,

Required probability is given by

   [Using R-code 'ppois(3,7)']

(e)

We observe that probabilities are almost same (0.08107379 and 0.08176542 and so differs by 0.00069163 only) in both cases i.e. using Binomial distribution and Poisson distribution. We know that Poisson approximation of Binomial distribution performs satisfactorily when number of trials is large (1056 here), probability of success is very small (7/1056 = 0.006628788 here) but expected number of success is finite (7 here). We observe that all of these three conditions are satisfied. Thus obtaining almost same probability in these two approaches is theoretically certain.