The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is _______.
Correct Answer: 96
Solution and Explanation
Given that,
np + npq = 82.5 … (1)
np (npq) = 1350 … (2)
Since, Mean and Vairance be the roots of x2 – 82.5x + 1350 = 0
⇒ x2 – 22.5 x – 60x + 1350 = 0
⇒ x – (x – 22.5) – 60 (x – 22.5) = 0
Mean = 60 and Variance = 22.5
np = 60, npq = 22.5
\(⇒q=\frac{9}{24}=\frac{3}{8},\)
\(⇒p=\frac{5}{8}\)
\(∴n\frac{5}{8}=60\)
\(⇒n=96\)
So, the answer is 96.
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Concepts Used:
Mean Value Theorem
The theorem states that for a curve f(x) passing through two given points (a, f(a)), (b, f(b)), there is at least one point (c, f(c)) on the curve where the tangent is parallel to the secant passing via the two given points is the mean value theorem.
The mean value theorem is derived herein calculus for a function f(x): [a, b] → R, such that the function is continuous and differentiable across an interval.
- The function f(x) = continuous across the interval [a, b].
- The function f(x) = differentiable across the interval (a, b).
- A point c exists in (a, b) such that f'(c) = [ f(b) - f(a) ] / (b - a)