Edexcel IAL - Statistics 2- 1.2 Mean and Variance of Binomial and Poisson Distributions- Study notes - New syllabus
Edexcel IAL – Statistics 2- 1.2 Mean and Variance of Binomial and Poisson Distributions -Study notes- New syllabus
Edexcel IAL – Statistics 2- 1.2 Mean and Variance of Binomial and Poisson Distributions -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 1.2 Mean and Variance of Binomial and Poisson Distributions
Mean and Variance of the Binomial and Poisson Distributions
For both the binomial and Poisson distributions, the mean and variance describe the central value and spread of the distribution. These results may be used directly; no derivations are required.
Binomial Distribution
If a random variable \( X \) has a binomial distribution with parameters \( n \) and \( p \), written as
\( X \sim \mathrm{Bin}(n,p) \)
then:
Mean: \( \mathrm{E}(X) = np \)
Variance: \( \mathrm{Var}(X) = np(1-p) \)
These depend on both the number of trials \( n \) and the probability of success \( p \).
Poisson Distribution
If a random variable \( X \) has a Poisson distribution with mean \( \lambda \), written as
\( X \sim \mathrm{Po}(\lambda) \)
then:
Mean: \( \mathrm{E}(X) = \lambda \)
Variance: \( \mathrm{Var}(X) = \lambda \)
A key feature of the Poisson distribution is that the mean and variance are equal.
Summary Table
| Distribution | Mean | Variance |
|---|---|---|
| Binomial \( \mathrm{Bin}(n,p) \) | \( np \) | \( np(1-p) \) |
| Poisson \( \mathrm{Po}(\lambda) \) | \( \lambda \) | \( \lambda \) |
Key Examination Points
- You must know and use these formulae accurately
- No proofs or derivations are expected
- Comparing mean and variance can help judge the suitability of a model
Example :
A random variable \( X \) has a binomial distribution with parameters \( n = 20 \) and \( p = 0.3 \).
Find the mean and variance of \( X \).
▶️ Answer/Explanation
Since \( X \sim \mathrm{Bin}(20, 0.3) \):
Mean: \( \mathrm{E}(X) = np = 20 \times 0.3 = 6 \)
Variance: \( \mathrm{Var}(X) = np(1-p) = 20 \times 0.3 \times 0.7 = 4.2 \)
Conclusion: The mean is 6 and the variance is 4.2.
Example :
The number of misprints per page in a book follows a Poisson distribution with mean 1.8.
State the mean and variance of the distribution.
▶️ Answer/Explanation
Let \( X \sim \mathrm{Po}(1.8) \).
Mean: \( \mathrm{E}(X) = \lambda = 1.8 \)
Variance: \( \mathrm{Var}(X) = \lambda = 1.8 \)
Conclusion: Both the mean and variance are equal to 1.8.
Example :
A random variable \( X \) has a Poisson distribution. It is known that the variance of \( X \) is 5.
Find the mean of \( X \). Hence state the distribution of \( X \).
▶️ Answer/Explanation
For a Poisson distribution:
\( \mathrm{Var}(X) = \lambda \)
Given \( \mathrm{Var}(X) = 5 \),
\( \lambda = 5 \)
Hence,
Mean: \( \mathrm{E}(X) = 5 \)
Conclusion: The distribution is \( X \sim \mathrm{Po}(5) \).