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) :