Question

1. A software program is able to recognize a hand written letter about 63% of the time.

(a) What is the probability that it takes less than 4 letters before the program recognizes one?

(b) What is the expected number of letters shown to the program before it recognizes one?

(c) If we show the program 20 letters, what is the probability that it recognizes exactly 12 of them?

(d) If we show the program 20 letters, what is the expected number of letters it recognizes, what is the variance?

(e) Suppose we show the program 5 sets of 20 letters each and record how many it regonizes each time. Use the maximum likelihood method to create an estimator for the program’s probability of recognizing a letter.

Answer 1

1.

(a)

We can model the situation using Geometric distribution as follows.

Suppose, random variable X denotes number of letters required to recognise first letter.

We define recognising a letter as success.

Recognising a letter is independent of recognising other letters.

Required probability is given by

(b)

We have

In case of geometric distribution,

Subtracting,

So, 1.59 letters are expected to be shown before recognising first one.

(c)

We can model the situation using Binomial distribution as follows.

Suppose, random variable Y denotes number of recognised letters.

We define recognising a letter as success.

Recognising a letter is independent of recognising other letters.

Required probability is given by

(d)

We have,

In case of Binomial distribution,

So, expected number of letter to recognise is 12.6 and variance is 4.662.

(e)

Suppose, be number of letters recognised in 5 sets.

For joint probability distribution function is given by

Differentiating with respect to p we get,

Equating L.H.S. to 0 we get,

Hence, maximum likelihood estimator for the program’s probability of recognising a letter is given by