Discrete and Continuous Random Variables Section 7.1

Random Variable  A random variable is a variable whose value is a numerical outcome of a random phenomenon.  An example of a random variable would be the count of heads in four coin tosses.  Random variables are usually denoted by capital letters near the end of the alphabet.

Discrete Random Variable  A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities: Value of X x 1 x 2 x 3 … x k Probability p 1 p 2 p 3 …. p k

 The probability p i must satisfy two requirements: 1. Every probability p i is a number between 0 and p 1 + p 2 +… + p k = 1.  Find the probability of any event by adding the probabilities p i of the particular values x i that make up the event.

Example  The instructor of a large class give 15% each of A’s and D’s, 30% each of B’s and C’s and 10% F’s. Choose a student at random from this class. The student’s grade on a four-point scale is a random variable X.  The value of X changes when we repeatedly choose students at random, but it is always 0, 1, 2, 3, or 4.

The probability distribution of X Value of X Probability The probability that the student got a B or better is the sum of the probabilities of an A and a B: P(grade is 3 or 4) = P(X=3) + P(X=4) = =.45

Probability Histogram  Can be used to picture the probability distribution of a discrete random variable.  A probability histogram is in effect a relative frequency histogram for a very large number of trials.

Example  What is the probability distribution of the discrete random variable X that counts the number of heads in four tosses of a coin?  Two assumptions must be made.  How many possible outcomes are there?  What is the probability of any one outcome?  What is the probability of tossing at least two heads?

Assignment  Page 470, problems 7.2 – 7.5

Continuous Random Variables  Choose a number between 0 and 1.  How many outcomes are possible?  How would you assign probabilities?  We use a new way of assigning probabilities – as areas under a density curve.  Any density curve has area exactly 1 underneath it, corresponding to a total probability of 1.

Definition  A continuous random variable X takes all values in an interval of numbers.  The probability distribution of X is described by a density curve.  The probability of any event is the area under the density curve and above the values of X that make up the event.  The probability model for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes.  All continuous probability distributions assign probability 0 to every individual outcome.

IMPORTANT!  We ignore the distinction between > and > when finding probabilities for continuous (but not discrete) random variables.

Normal distributions as probability distributions  The density curves most familiar to us are the normal curves.  Normal distributions are probability distributions.  Example 7.4 page 474

Assignment  Page 379, problems 7.7 – 7.9

Section Summary  Random Variable  Probability distribution  Discrete random variable  Continuous random variable  Density curve  Normal distributions  Probability histogram

Assignment  Page 477, problems 7.11 – 7.17, 7.20