# AP Statistics Curriculum 2007 Limits Norm2Poisson

(Difference between revisions)
 Revision as of 01:53, 3 February 2008 (view source)IvoDinov (Talk | contribs) (→Normal Approximation to Poisson Distribution)← Older edit Revision as of 01:58, 3 February 2008 (view source)IvoDinov (Talk | contribs) (→Normal Approximation to Poisson Distribution)Newer edit → Line 4: Line 4: The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson($\lambda$) distribution]] can be approximated with [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal]] when $\lambda$ is large. The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson($\lambda$) distribution]] can be approximated with [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal]] when $\lambda$ is large. - For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal($\mu=\lambda, \sigma^2=\lambda$]] distribution is an excellent approximation to the Poisson distribution.  If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate [[continuity correction]] is performed, i.e., P(''X'' ≤ ''x''), where (lower-case) ''x'' is a non-negative integer, is replaced by P(''X'' ≤ ''x'' + 0.5). + For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal($\mu=\lambda, \sigma^2=\lambda$)]] distribution is an excellent approximation to the [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(λ)]] distribution.  If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. Suppose P(''X''≤''x''), where (lower-case) ''x'' is a non-negative integer, is replaced by P(''X'' ≤ ''x'' + 0.5). ::  $F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,$ ::  $F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,$ + If $X\sim Poisson(\lambda)$ and $U\sim Normal([itex]\mu=\lambda, \sigma^2=\lambda$), then [itex]P_X(X

## General Advance-Placement (AP) Statistics Curriculum - Normal Approximation to Poisson Distribution

### Normal Approximation to Poisson Distribution

The Poisson(λ) distribution can be approximated with Normal when λ is large.

For sufficiently large values of λ, (say λ>1,000), the Normal(μ = λ,σ2 = λ) distribution is an excellent approximation to the Poisson(λ) distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed. Suppose P(Xx), where (lower-case) x is a non-negative integer, is replaced by P(X ≤ x + 0.5).

$F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,$

If $X\sim Poisson(\lambda)$ and $U\sim Normal([itex]\mu=\lambda, \sigma^2=\lambda$), then PX(X < xo) = PU(U < xo + 0.5).

### Examples

Suppose cars arrive at a parking lot at a rate of 50 per hour. Let’s assume that the process is a Poisson random variable with λ = 50. Compute the probability that in the next hour the number of cars that arrive at this parking lot will be between 54 and 62. We can compute this as follows: $P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617.$ The figure below from SOCR shows this probability.

• Note: We observe that this distribution is bell-shaped. We can use the normal distribution to approximate this probability. Using $N(\mu=50, \sigma=\sqrt{50}=7.071)$, together with the continuity correction for better approximation we obtain $P(54 \le X \le 62)=0.2718$, which is close to the exact that was found earlier. The figure below shows this probability.