Probability
Binomial distribution, mean and variance
nta_pyq_2023_jan
Grade None
Question:
In a binomial distribution $B(n, p)$, the sum and product of the mean and variance are 5 and 6 respectively, then find $6(n + p - q)$ is equal to:
Step-by-Step Solution
Key Concept: Mean $= np$, variance $= npq$; set up $np + npq = 5$ and $np \cdot npq = 6$, then solve for $n, p, q$
$np(1+q)=5$, $n^2p^2q=6$. From these: $6(1+q)^2/q = 25 \Rightarrow 6q^2-13q+6=0 \Rightarrow q=2/3$ (taking valid root). $p=1/3$. $n(1/3)(5/3)=5 \Rightarrow n=9$. $6(n+p-q) = 6(9+1/3-2/3)=6(9-1/3)=6(26/3)=52$. Answer: (2)
Correct Answer: 52