Question

Find the 30th and the 75th percentiles of BIN(n=11,p=0.5).

(My thought process is to approximate it to Normal Distribution and approximate the percentile from there, is it the right way or there is a way without approximation?) Please show your workings.

Answer 1

The 30th percentile would be that value of r (number of successes) where the cumulative probability of that event happening is 0.30. Similarly, the 75th percentile would be that value of r where the cumulative probability of the event happening is 0.75.

This would be easy to solve substituting the value of r from 0 till we reach a cumulative probability of 0.70.

We know that the PMF of a binomial distribution is ⁿC_r * p^r * (1-p)^n-r.

Substitute the value of r starting from 0.

Thus, when r is 0, we get, ¹¹C₀ * 0.5⁰ * (0.5)¹¹ = 0.00048828125

Similarly, when r is 1, we get 0.00537109375.

When r is 2, we get 0.02685546875.

When r is 3, we get 0.08056640625.

When r is 4, we get 0.1611328125.

When r is 5, we get 0.2255859375.

When r is 6, we get 0.2255859375.

When r is 7, we get 0.1611328125.

When r is 8, we get 0.08056640625.

Let us make a table of the probabilities and the cumulative probabilities.

x	P(X=x)	P(X<x)
0	0.00048828125	0.00048828125
1	0.00537109375	0.005859375
2	0.02685546875	0.03271484375
3	0.08056640625	0.11328125
4	0.1611328125	0.2744140625
5	0.2255859375	0.5
6	0.2255859375	0.7255859375
7	0.1611328125	0.88671875
8	0.08056640625	0.96728515625

From this, we can see that the 30th percentile, or the value of r where the cumulative probability is less than 30% is 5.

Similarly, the 75th percentile would be 7.

Please let me know (in the comments) if you have a doubt. Happy learning!