ST 351

Continuous Random Variables

 

 

A second type of random variable is one that has a continuous distribution.  (That is, the random variable is continuous.)  We’ll refer to these as continuous random variables. A continuous random variable can assume an infinite number of possible values within a certain interval.

 

A continuous random variable can be described by a density curve.  Recall that a density curve has two properties:

            1)         it lies on or above the x-axis

            2)         the area under the density curve and above the x-axis equals one.

 

This second property is equivalent to saying that the sum of the probabilities of all of the possible outcomes in the sample space is 1.  For a continuous random variable, the sample space has an infinite number of possible outcomes.  But, altogether, the probability of the sample space is 1.     

 

Therefore, finding the probability of a certain event occurring can be determined by finding a certain area under the curve corresponding to that event.  As an example, let’s consider one type of continuous random variable that has a uniform distribution.  (A continuous random variable that has a uniform distribution means that if a histogram were drawn, each “class” would contain the same number of observations – evenly distributed across the range of the data.)

 

Uniform distribution

 

The following is the density curve from a uniform random variable, X, which could assume any value between 0 and 2.  Find the following probabilities:

 

 

Find the following probabilities:

 

1)  P(.2<X1.4)             2)  P(X=1)                      3)  P(X>0.5)                   4)  P(X 1.2)

 

 

 

 

 

 

Another type of continuous random variable is one that has a normal distribution.  Note that a continuous random variable can have many other types of distributions besides a uniform distribution and a normal distribution, but these are the only two we will talk about.

 

Normal distribution

 

The psychology department at a university finds that the scores of its applicants on the quantitative GRE are approximately normally distributed with mean  and standard deviation . 

 

1.         Sketch this distribution.  Label the x-axis and identify the mean on the x-axis.  Also identify the values on the x-axis one standard deviation greater than and less than the mean.

 

           

 

 

2.         Suppose one person is chosen at random from this population who had a GRE score of 647.  How many standard deviations away from the mean is this person’s score?

 

 

 

 

 

 

 

2.         Same question and information as number 1, except the person’s score is 700?

 

 

 

 

 

 

3.         Again, same as number one, except the person’s score is 350?

 

 

 

 

 

 

 

 

 

 

 

 

Steps in finding probabilities for data that’s normally distributed:

 

    1.     Draw a normal curve.  On the x-axis, identify the mean, .  Also on the x-axis, mark the particular value of X in which you are interested.  Draw a vertical line at this point from the x-axis to the density curve.  Shade the area under the normal curve that corresponds to the probability you want to find.  This probability is in terms of X:  P(X).

    2.     Calculate the z-score (also referred to as the z-statistic).  This is the number of standard deviations X is away from .

 

                                          where X is the observed value

 

    3.     Draw the Standard Normal Curve.  (The standard normal curve has z-statistics on the x-axis.) The mean is now zero.  The value of X in which you are interested is now Z.  Again, shade the area corresponding to the probability you want to find.  Now, the probability is in terms of Z:  P(Z)

    4.     Use the table (Table A or inside the front cover) to find the corresponding probability.  Remember, the table gives P(Z<z-statistic)

 

            what you want                           picture                                      ­finding prob. using table

            P(Z<z-statistic)                                                                          take the probability from the

Ex:  P(Z<1)                               table

                       

 

            P(Z>z-statistic)                                                                          1 – probability from the table

                                                Ex: P(Z>1)

                       

 

 

 

P(z-stat#1<Z<z-stat#2)                                                               probability from the table for

Ex: P(-1<Z<1)                                        z-stat#2 – probability from the table for z-stat#1

 

                       

 

4.         Suppose one person is chosen at random from this population.  What is the probability this person scores between 350 and 700?

 

 

 

 

 

 

 

 

 

5.         What score does this person need to have in order for he/she to be in the top 10% of all scores?

 

 

 

 

 

 

 

 

6.         Suppose 300 people took the GRE.  How many would you expect to score above 700?