4.17 Poisson distributionIB Maths: Applications and Interpretation HL: Revision notes

The Poisson distribution models the number of events $X$ occurring in a fixed interval of time or space. It is appropriate when:

• events are independent of one another, and
• events occur at a uniform average rate (constant over the period of interest), and can occur at any point in the interval. We write $X\sim\mathrm{Po}(\lambda)$, where $\lambda>0$ is the mean number of events in the interval. Examples: calls to an office per hour, flaws per square metre of fabric, typing errors per page. If events come in clusters (for example several patients from one accident), or the rate changes through the day, the model is not appropriate.

For $X\sim\mathrm{Po}(\lambda)$: $\mathrm{P}(X=r)=\frac{e^{-\lambda}\lambda^r}{r!},\quad r=0,1,2,\dots$ $\mathrm{E}(X)=\lambda,\qquad\mathrm{Var}(X)=\lambda.$ The mean and variance are equal. Use the GDC (Poisson pdf for $\mathrm{P}(X=r)$, Poisson cdf for $\mathrm{P}(X\le r)$). Example: $\lambda=4$: $\mathrm{P}(X=3)=0.195$, $\mathrm{P}(X\le3)=0.433$. For $\mathrm{P}(X\ge r)$ use $1-\mathrm{P}(X\le r-1)$; so $\mathrm{P}(X>3)=1-\mathrm{P}(X\le3)$. Formal proofs of the mean and variance are not required.

The parameter $\lambda$ is the mean for the interval in the question. If the rate is $4$ per hour then for $30$ minutes $\lambda=2$, and for three hours $\lambda=12$ (so the variance is also $12$). Always rescale $\lambda$ first, then use the distribution for the new interval. Example: patients arrive at $5$ per hour. In two hours $Y\sim\mathrm{Po}(10)$ and $\mathrm{P}(Y>12)=1-\mathrm{P}(Y\le12)=0.208$.

If $X\sim\mathrm{Po}(\lambda)$ and $Y\sim\mathrm{Po}(\mu)$ are independent, then $X+Y\sim\mathrm{Po}(\lambda+\mu).$ Example: flaws at $0.8$ and $1.5$ per square metre; one square metre of each gives $\mathrm{Po}(2.3)$ and $\mathrm{P}(\text{none})=e^{-2.3}=0.100$. For joint events use independence: $\mathrm{P}(X=0\text{ and }Y=2)=\mathrm{P}(X=0)\,\mathrm{P}(Y=2)$.

• Binomial $\mathrm{B}(n,p)$: a fixed number $n$ of independent trials, each success or failure with constant probability $p$ (a count out of $n$).
• Poisson $\mathrm{Po}(\lambda)$: events occurring at random in a continuous interval of time or space, at a constant average rate, with no upper limit on the count.
• Normal $N(\mu,\sigma^2)$: a continuous measurement (mass, time, height) that is symmetric about the mean. Check the context: 'out of $80$ bottles, each cracked with probability $0.03$' is binomial; '$0.5$ bubbles per bottle on average' is Poisson; the mass of a loaf is normal.