Section 6.3 Geometric Random Variables

Binomial and Geometric Random Variables Geometric Settings In a binomial setting, the number of trials n is fixed and the binomial random variable X counts the number of successes. In other situations, the goal is to repeat a chance behavior until a success occurs. These situations are called geometric settings. Definition: A geometric setting arises when we perform independent trials of the same chance process and record the number of trials until a particular outcome occurs. The four conditions for a geometric setting are Binary? The possible outcomes of each trial can be classified as “success” or “failure.” Independent? Trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. Trials? The goal is to count the number of trials until the first success occurs. Success? On each trial, the probability p of success must be the same. B I T S

Comparison of Binomial to Geometric BinomialGeometric Each observation has two outcomes (success or failure). The probability of success is the same for each observation. The observations are all independent. There are a fixed number of trials. There is a fixed number of successes (1). So, the random variable is how many successes you get in n trials. So, the random variable is how many trials it takes to get one success.

How to Calculate Geometric Probabilities It’s usually not difficult to calculate these by hand. It’s usually not difficult to calculate these by hand. Let’s revisit Chris’ free throw shooting. Let’s revisit Chris’ free throw shooting. Currently, he is a 75% free throw shooter. Currently, he is a 75% free throw shooter. What is the P(X = 3)? That means what is the probability that he makes his first basket on the third shot? What is the P(X = 3)? That means what is the probability that he makes his first basket on the third shot?

Let’s Construct a Probability Distribution for a Geometric Random Variable Suppose Coach Roth helps Chris improve to be a 80% free throw shooter. Suppose Coach Roth helps Chris improve to be a 80% free throw shooter. Begin constructing a probability distribution for how many shots it takes for him to make his first free throw. Begin constructing a probability distribution for how many shots it takes for him to make his first free throw. What would the graph of a geometric probability distribution look like? What would the graph of a geometric probability distribution look like?

The Probability Distribution for the Geometric R.V. Calculating a > or a ≥ probability should use the converse rule. Calculating a > or a ≥ probability should use the converse rule. So, P(X > 6) = 1 – P(X ≤ 6) So, P(X > 6) = 1 – P(X ≤ 6) Using the “New Chris” example, find the following probabilities. Using the “New Chris” example, find the following probabilities. P(X<3) P(X≥5) P(X<3) P(X≥5)

Mean & Variance of Geometric RV The formula for the mean of a geometric RV is The formula for the variance of a geometric RV is These formulas are NOT given to you on the exam.

Mean & Variance of Geometric RV Let’s revisit “New Chris” one more time. Let’s revisit “New Chris” one more time. Let X = when Chris ______________ Let X = when Chris ______________ What is the expected value of X? What is the expected value of X? What is the standard deviation of X? What is the standard deviation of X?

Putting it all together… We’ve studied two large categories of RVs: discrete and continuous Among the discrete RVs, we’ve studied the binomial and geometric Among the discrete RVs, we’ve studied the binomial and geometric The graph of a binomial RV can be skewed left, symmetric, or skewed right, depending on the value of p. The graph of a binomial RV can be skewed left, symmetric, or skewed right, depending on the value of p. The graph of a geometric RV is ALWAYS skewed right. Always. The graph of a geometric RV is ALWAYS skewed right. Always. Other discrete RVs can be given to you in the form of a table. Other discrete RVs can be given to you in the form of a table. Among the continuous, we’ve studied the normal RVs. Among the continuous, we’ve studied the normal RVs. To find probabilities of a normal RV, convert to a Z score and use Table A. To find probabilities of a normal RV, convert to a Z score and use Table A.