Plots of Poisson Distribution

Example

When can you use the Normal distribution to approximate the Poisson distribution, assuming you have "n" independent trials each with a probability of success of "p"?

1. When the mean of the Poisson distribution is very small
2. When the variance of the Poisson distribution is very small
3. When the number of observations is very large and the success rate is close to 1
4. When the number of observations is very large and the success rate is close to 0

Solution

C.
The Normal distribution can approximate the distribution of a Poisson random variable with a large lambda parameter (Î»).  This will be the case when both the number observations (n) is very large and the success rate (p) is close to 1 since Î» = n*p.
INCORRECT: A, The mean of a Poisson distribution must be large to allow approximation with a Normal distribution
INCORRECT: B, The variance of a Poisson distribution must be large to allow approximation with a Normal distribution
INCORRECT: D, The Normal distribution can approximate the distribution of a Poisson random variable with a large lambda parameter (Î»).  But since Î» = n*p, where n is the number observations and p is the success rate, Î» will not be large if p is close to 0

Example

The number of false fire alarms in a suburb of Houston averages 2.1 per day. What is the (approximate) probability that there would be 4 false alarms on 1 day?

1. 1.0
2. 0.1
3. 0.5
4. 0.0

Solution

B.
Use Poisson distribution
P(X = x) = [Î» x *e-Î»]/ Î»!
Is there any other intuitive way as well???