Normal Approximation to the Binomial

This applet illustrates the normal approximation for the binomial distribution.

The important points to remember and notice on the program in regard to the binomial are:

• Probabilities for the binomial are defined for integer values. As such, the binomial is said to be a probability mass function as the probabilities are concentrated (in mass) at distinct points.
• The cumulative probability function for the binomial is a step function. It has jumps at each integer where the probability mass changes.
• The binomial is characterized by the sample size n and the probability of success p. It has mean np and variance np(1-p).

The important points to remember and notice on the program in regard to the normal are:

• Probabilities for the normal are defined at all values. As such, the normal is said to be a probability density function as probabilities are calculated for ranges of points. Since the outcome is continuous, the probability of observing any distinct value is zero.
• The cumulative probability function for the normal is a smooth curve that continuously changes as the range gets larger (and thus the probability changes).
• In order to obtain approximations to probabilities for the discrete binomial, we must choose some reasonable interval for which to calculate probabilities with the normal. We use an interval of length 1 centered at the value of interest.
• The normal is characterized by the mean u and the variance s². The approximating normal (to the binomial) has mean u=np and variance s²=np(1-p).