~Continuous Probability Distributions:  There are many...some very complicated, but all are based on an infinite amount of continuous data. Think of the data as intervals on the x-axis & the y-value, for each x-value, as its frequency.

~They are defined by a connected (continuous) curve called a probability density function that have the very important properties of: (1) the curve is always nonnegative & (2) the total area under the curve is exactly equal to 1. These requirements are necessary for all probability density functions.

~We can then make a tie between probability & the area under the curve (calculus based).

~Note: if X represents our continuous random variable, then the probability X takes on one particular value is zero.  (the area under a continuous curve above one point on the X axis is zero). So, we only consider the probability that X lies between two different values.

~The Uniform Probability Distribution:  this is the most simplest type & have the same frequency for all values of X. The probability density function (curve) will look horizontal. So, in this case, the area under the curve & between two values of X can easily be found by inspection. Which means, that the probabilities can be found easily. (areas & probabilities are one of the same).

~Note: since all areas under this curve are rectangles, the probabilities can be found easily.

~Note: the most important types of probability density functions are the Normal Distributions.

~There are many & look bell-shaped. Their means, standard deviations, & heights (kurtosis) could vary.

~Note: the equation for this curve is quite complicated & is not given here.

~Note: As sample sizes increase, the means of most populations being sampled approach a normal distribution regardless of whether on not the population itself is considered to be normally distributed or not. This very important property is know as the central limit theorem & will be covered shortly.              

~How do you find Areas (probabilities)?  Convert the normal curve to the Standard Normal Curve.

~the standard normal curve (also called the z-curve) has mean,
m = 0 & standard deviation, s = 1

~Note: the horizontal axis contains standard deviation values.
           These are known as z-scores.
           So, it’s labeled the z-axis.

~Note that where the X=m, z=0. The formula connecting the X score & the z-score is  Z = (X - m)/s & must be memorized for future work.

~popular z values that give important areas are, as follows:
From z = -1 to z = +1, the area under the curve is .6826
From z = -2 to z = +2 , the area under the curve is .9544
From z = -3 to z = +2 , the area under the curve is .9974

~Finding probabilities (areas) from z-scores using Table A-2
(see inside book cover) or, better yet, use your TI-83.(see below)

~Ex:  For z = 1.83.  Should get .9664

~Ex:  Find area between z = -1.45 and z = 0.   
         Should get .4265

~Ex:  Find the probability a score is to the right of z = 1.25.  
          Should get .1056

~Ex:  Find the probability a score is between z = .46 and z = 1.75.     
         Should get .2827

~Going in reverse direction: given an area (probability), find the z-score

~your probability (area) may not be listed in table A-2, so you’ll have to estimate the z-score or, better yet, use your TI-83.(see below)

~Note:  We will use the TI-83 to do both. Here are the menus:

1) finding probabilities (areas) using z scores:  
    Use  2nd VARS, to 2 (normal cdf), enter the two z scores separated   
    by comma, then enter.(gives an area) (probability)

~Note:  for an area greater (to the right) of a given z score, enter   
             10,000 for the 2nd z score.
              For an area less than (to the left) of a given z score, enter
              -10,000 for the 1st z score.

2) finding a z score given an area (probability):.
   Use  2nd VARS, to 3 (invNorm), enter the area (probability), enter
   (gives z score)

~Note: These techniques make it very easy to find percentile scores, since a percentile score is where a certain percentage of the scores are below that score (an area or probability). Use procedure 2).

~Note: If X scores are known & you wish to find probabilities (areas) related to them, you can avoid finding their Z scores simply by using 2nd VARS, menu 2 & inserting (X score to the left, X score to the right, the mean, the standard deviation).

~likewise, you can get an X score, if the area to the left of the score is known, by using menu 3 & inserting (area or probability, the mean, the standard deviation). This is very useful for finding pecentile scores.