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A binomial distribution with parameters m and p can be approximated by a Poisson distribution with parameter m = np is
- a)m → ∝.
- b)p → 0.
- c)m → ∝ and p → 0.
- d)m → ∝ and p → 0 so that mp remainsfinite..
Correct answer is option 'D'. Can you explain this answer?
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A binomial distribution with parameters m and p can be approximated by...
Binomial Distribution and Poisson Distribution
Binomial Distribution:
The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. Let X be the number of successes in n trials, then X has a binomial distribution with parameters n and p, denoted by X~B(n,p)
Poisson Distribution:
The Poisson distribution is a discrete probability distribution that describes the number of occurrences of a rare event within a fixed interval of time or space. Let X be the number of occurrences of the event in the interval, then X has a Poisson distribution with parameter λ, denoted by X~Pois(λ).
Approximation of Binomial Distribution by Poisson Distribution
When n is large and p is small, it is often difficult and time-consuming to compute binomial probabilities. In such cases, we can approximate the binomial distribution by a Poisson distribution with parameter λ = np.
The conditions under which the binomial distribution can be approximated by a Poisson distribution are:
1. n is large
2. p is small
3. np is finite
In such cases, the Poisson distribution provides a good approximation to the binomial distribution.
Applying these conditions to the given options:
a) Incorrect: Only m is given, we cannot conclude anything about p.
b) Incorrect: Only p is given, we cannot conclude anything about m.
c) Incorrect: Only m is given, we cannot conclude anything about p.
d) Correct: The condition np is finite is satisfied. Therefore, we can approximate the binomial distribution with parameters m and p by a Poisson distribution with parameter λ = np = mp.
Binomial Distribution:
The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. Let X be the number of successes in n trials, then X has a binomial distribution with parameters n and p, denoted by X~B(n,p)
Poisson Distribution:
The Poisson distribution is a discrete probability distribution that describes the number of occurrences of a rare event within a fixed interval of time or space. Let X be the number of occurrences of the event in the interval, then X has a Poisson distribution with parameter λ, denoted by X~Pois(λ).
Approximation of Binomial Distribution by Poisson Distribution
When n is large and p is small, it is often difficult and time-consuming to compute binomial probabilities. In such cases, we can approximate the binomial distribution by a Poisson distribution with parameter λ = np.
The conditions under which the binomial distribution can be approximated by a Poisson distribution are:
1. n is large
2. p is small
3. np is finite
In such cases, the Poisson distribution provides a good approximation to the binomial distribution.
Applying these conditions to the given options:
a) Incorrect: Only m is given, we cannot conclude anything about p.
b) Incorrect: Only p is given, we cannot conclude anything about m.
c) Incorrect: Only m is given, we cannot conclude anything about p.
d) Correct: The condition np is finite is satisfied. Therefore, we can approximate the binomial distribution with parameters m and p by a Poisson distribution with parameter λ = np = mp.
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A binomial distribution with parameters m and p can be approximated by a Poisson distribution with parameter m = np isa)m .b)p 0.c)m and p 0.d)m and p 0 so that mp remainsfinite..Correct answer is option 'D'. Can you explain this answer?
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