Some Useful Continuous Probability Distributions

8.1 Properties of Continuous Probability Distributions A smooth curve known as the density function, is used to represent the probability distribution of a continuous random variable The curve must never fall below the x-axis … for all x The total area under the curve must be 1 …

For continuous random variables we assign probability to intervals. (Not Points) the area under the curve between a and b With continuous variables, each point has probability zero Thus for continuous variables

For continuous distributions Population Mean = Population Variance = Population Standard Deviation =

8.2 The Uniform Distribution The density function for the uniform distribution is as follows: for Calculating descriptive statistics Population Mean = Population variance = Population standard deviation =

The probability that a value is between c and d is where Example: Travel time from Lexington KY to Columbus OH is a uniform distributed between 200 and 240 minutes Give the density function Find the mean. Find the median. Find the variance. Find the standard deviation. Find the probability of arriving in less than 225 minutes.

8.3 The Normal Distribution Density function for the normal distribution is as follows:

The normal distribution is a common type of continuous distribution. It is a bell shaped curve. The bell is symmetric about the mean of the random variable. The standard deviation of the random variable measures the spread of the bell. The larger is the more spread out the bell. For the normal the mean, median, and mode are equal. The value of and characterize which normal distribution we are using

The normal distribution with and is called the standard normal distribution. (This is used to calculate probabilities for all normal distributions) If X is normal with mean and standard deviation, then is standard normal.

Examples Draw Pictures of desired areas when doing problems!!! Facts: Total area under the curve is 1 Curve is symmetric about 0 Combining these facts with the table allows us to compute all probability statements for Z

Examples Notice that Probabilities in the table stop at 3.9 with an area of Beyond this Z value you will always have close to.5 the area.

8.4 Calculating Areas Under Any Normal Curve If x is normal with mean and standard deviation, then is standard normal. Write probability statement for X Rewrite in terms of Z

Example The distribution of IQ scores for the general population is approximately normal with and. x = IQ score of randomly selected person Find

Example Suppose the amount of Pepsi in a “12 oz” can has a Normal distribution with and X = amount of Pepsi in a Randomly selected can Find