# AP Statistics Curriculum 2007 Limits Norm2Poisson

(Difference between revisions)
 Revision as of 01:57, 3 February 2008 (view source)IvoDinov (Talk | contribs) (→Normal Approximation to Poisson Distribution)← Older edit Revision as of 22:12, 1 March 2008 (view source) (→Normal Approximation to Poisson Distribution)Newer edit → Line 2: Line 2: === Normal Approximation to Poisson Distribution=== === Normal Approximation to Poisson Distribution=== - 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 [[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. + 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. If $x_o$ is a non-negative integer, $X\sim Poisson(\lambda)$ and $U\sim Normal(\mu=\lambda, \sigma^2=\lambda$), 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.

If xo is a non-negative integer, $X\sim Poisson(\lambda)$ and $U\sim Normal(\mu=\lambda, \sigma^2=\lambda$), 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.