UPSC NDA I 2024 Mathematics Paper

A random variable x that follows a binomial distribution can be expressed as X∼Binomial(n,p). The mean of a binomial distribution is given by np and the variance is np(1−p).
We are given that the mean (np) is 5 and the variance (np(1−p)) is 4 . Therefore, we have two equations:
np=5
np(1−p)=4
We can use the first equation to express p in terms of n :
p=‌

5

n

Substituting p in the second equation:
np(1−‌

5

n

)=4
5(1−‌

5

n

)=4
5−‌

25

n

=4
1=‌

25

n

n=25
So, n=25 and p=‌

5

25

=‌

1

5

.
Now, we need to find λ from the given equation 523P(X=3)=λ4λ. We use the binomial probability formula to find P(X=3) :

P(X=3)=(

25

3

)(‌

1

5

)3(‌

4

5

)22
P(X=3)=‌

25!

3!⋅22!

(‌

1

125

)(‌

4

5

)22
P(X=3)=2300⋅‌

1

125

⋅(‌

4

5

)22
P(X=3)=18.4⋅(‌

4

5

)22
Therefore,
523P(X=3)=523×18.4×(‌

4

5

)22
=5×18.4×422∕522
=18.4×‌

422

521

This can be written as:
λ4λ=18.4×‌

422

521

By comparison, it is clear that λ=23.