Chapter 6 – Continuous Probability Distribution Introduction A probability distribution is obtained when probability values are assigned to all possible numerical values of a random variable. It may also be denoted by the symbol f(x), in the continuous, which indicates that a mathematical function is involved. The sum of the probabilities for all the possible numerical events must equal 1.0.

6.1 Normal Probability Distribution Definition 6.1

6.2 Standard Normal Probability Distribution The normal distribution with parameters and is called a standard normal distribution. A random variable that has a standard normal distribution is called a standard normal random variable and will be denoted by.

Standardizing A Normal Distribution

Example 6.1 Determine the probability or area for the portions of the Normal distribution described.

Solutions:

d) e)

Example 6.2

Solutions:

Example 6.3 Suppose X is a normal distribution N(25,25). Find

Solutions:

6.3 Normal Approximation of the Binomial Distribution  When the number of observations or trials (n) in a binomial experiment is relatively large, the normal probability distribution can be used to approximate binomial probabilities. A convenient rule is that such approximation is acceptable when  Definition 6.2

Continuous Correction Factor The continuous correction factor needs to be made when a continuous curve is being used to approximate discrete probability distributions. 0.5 is added or subtracted as a continuous correction factor according to the form of the probability statement as follows:

Example 6.4 In a certain country, 45% of registered voters are male. If 300 registered voters from that country are selected at random, find the probability that at least 155 are males.

Solutions:

6.4 Normal Approximation of the Poisson Distribution  When the mean of a Poisson distribution is relatively large, the normal probability distribution can be used to approximate Poisson probabilities. A convenient rule is that such approximation is acceptable when  Definition 6.3

Example 6.5 A grocery store has an ATM machine inside. An average of 5 customers per hour comes to use the machine. What is the probability that more than 30 customers come to use the machine between 8.00 am and 5.00 pm?

Solutions