AcademyProbability
Academy
Normal Approximation
Level 1 - Math II (Physics) topic page in Probability.
Principle
Normal distributions can approximate cumulative probabilities for some discrete count distributions. The approximation is most useful for ranges of counts, and a continuity correction accounts for the fact that integer counts are being represented by intervals on a continuous scale.
Notation
\(X\sim\operatorname{Bin}(n,p)\)
binomial count with n trials and success probability p
\(Y\sim\operatorname{Po}(\lambda)\)
Poisson count with parameter lambda
\(\mu\)
mean of the approximating normal distribution
\(\sigma^2\)
variance of the approximating normal distribution
\(Z\)
standard normal variable after standardisation
\(\Phi\)
standard normal cumulative distribution function
Method
Step 1: Match the count distribution mean and variance
For a binomial count, use \(\mu=np\) and \(\sigma^2=np(1-p)\). For a Poisson count, use \(\mu=\lambda\) and \(\sigma^2=\lambda\).
Binomial approximation
\[X\sim\operatorname{Bin}(n,p)\approx N(np,np(1-p))\]
Poisson approximation
\[Y\sim\operatorname{Po}(\lambda)\approx N(\lambda,\lambda)\]
Step 2: Apply a continuity correction for integer counts
An event such as \(X\gt40\) means \(X\ge41\) for an integer count. On the continuous normal scale, approximate this with the interval above \(40.5\).
Upper-tail count
\[P(X\gt40)=P(X\ge41)\]
Continuity correction
\[P(X\ge41)\approx P(N\gt40.5)\]
Standardise
\[P(N\gt40.5)=P\left(Z\gt\frac{40.5-\mu}{\sigma}\right)\]
Rules
Binomial mean
\[\mu=np\]
Binomial variance
\[\sigma^2=np(1-p)\]
Binomial guide
\[np\ge10\quad\text{and}\quad np(1-p)\ge10\]
Poisson mean and variance
\[\mu=\lambda,\quad \sigma^2=\lambda\]
Poisson guide
\[\lambda\ge5\]
Standardisation
\[Z=\frac{x-\mu}{\sigma}\]
Examples
Question
A factory makes
\[1500\]
gadgets, each defective with probability \[0.02\]
Approximate the probability of more than \[40\]
defects.Answer
For
\[X\sim\operatorname{Bin}(1500,0.02)\]
\[\mu=np=30\]
and \[\sigma^2=np(1-p)=29.4\]
so \[\sigma=\sqrt{29.4}\]
More than \[40\]
means at least \[41\]
so use \[P(X\gt40)\approx P\left(Z\gt\frac{40.5-30}{\sqrt{29.4}}\right).\]
Checks
- Use the approximation for cumulative probabilities such as tails and intervals.
- Do not treat exact point probabilities as normal areas without care.
- For binomial counts, check both \(np\ge10\) and \(np(1-p)\ge10\).
- For Poisson counts, check \(\lambda\ge5\).
- Use continuity corrections because integer counts are approximated by a continuous variable.