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The maximum value of the variance of a binomial distribution with parameters n and p is
- a)n/2.
- b)n/4.
- c)np (1 – p).
- d)2n.
Correct answer is option 'B'. Can you explain this answer?
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The maximum value of the variance of a binomial distribution with para...
Explanation:
Binomial Distribution: A binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions. The binomial distribution formula is:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
Where,
P(X=k) is the probability of getting ‘k’ successes in ‘n’ trials,
C(n,k) is the number of possible combinations of ‘n’ trials where ‘k’ successes are observed,
p is the probability of success in any given trial,
(1-p) is the probability of failure in any given trial,
n is the total number of trials.
Variance of Binomial Distribution: The variance of a binomial distribution is given by:
Var(X) = np(1-p)
Where,
n is the number of trials,
p is the probability of success in any given trial.
Maximum Variance of Binomial Distribution: To find the maximum value of the variance of a binomial distribution, we take the first derivative of the variance with respect to p and equate it to zero. This gives us the maximum value of the variance as:
p = 1/2
Substituting this value of p in the variance formula, we get:
Var(X) = n/4
Therefore, the maximum value of the variance of a binomial distribution with parameters n and p is n/4.
Answer: The correct answer is option 'B'.
Binomial Distribution: A binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions. The binomial distribution formula is:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
Where,
P(X=k) is the probability of getting ‘k’ successes in ‘n’ trials,
C(n,k) is the number of possible combinations of ‘n’ trials where ‘k’ successes are observed,
p is the probability of success in any given trial,
(1-p) is the probability of failure in any given trial,
n is the total number of trials.
Variance of Binomial Distribution: The variance of a binomial distribution is given by:
Var(X) = np(1-p)
Where,
n is the number of trials,
p is the probability of success in any given trial.
Maximum Variance of Binomial Distribution: To find the maximum value of the variance of a binomial distribution, we take the first derivative of the variance with respect to p and equate it to zero. This gives us the maximum value of the variance as:
p = 1/2
Substituting this value of p in the variance formula, we get:
Var(X) = n/4
Therefore, the maximum value of the variance of a binomial distribution with parameters n and p is n/4.
Answer: The correct answer is option 'B'.
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Community Answer
The maximum value of the variance of a binomial distribution with para...
Variance = np(1-p)
= n×1/2*1/2 [ since max value for p and q is 1/2]
=n/4
= n×1/2*1/2 [ since max value for p and q is 1/2]
=n/4
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The maximum value of the variance of a binomial distribution with parameters n and p isa)n/2.b)n/4.c)np (1 – p).d)2n.Correct answer is option 'B'. Can you explain this answer?
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