Section 1 – Discrete and Continuous Random Variables AP Statistics Chapter 6 Section 1 – Discrete and Continuous Random Variables

Random Variables A variable that takes a numerical value that describes the outcome of a chance process Example: You toss a fair coin 3 times Let x= the number of heads HHH HHT HTH THH HTT THT TTH TTT Value of x 1 2 3 Probability 1/8 3/8

Probability Distribution Previous table shows distribution of probabilities for x. Probability distribution of a random variable gives the probability for each possible value of the variable. The example is a Discrete Random Variable meaning there are a finite number of definable possible outcomes.

Probability Distribution of Discrete Random Variables Every probability must be between 0 and 1 The sum of all probabilities in distribution must =1. To find the probability of any event involving x, add all the probabilities xi that make up the event.

Apgar Scores A system of rating a babies health at birth (because the newborn must be classified!) 5 criteria rated 0-2 and summed for a total score from 0-10 7 or higher indicates a healthy baby What is the probability a randomly selected baby is healthy? Apgar Value 1 2 3 4 5 6 7 8 9 10 Prob. .001 .006 .007 .008 .012 .020 .038 .099 .319 .437 .053 Is this a valid probability distribution?

Mean of a Discrete Random Variable Just as in analyzing data, we sometimes want to use a statistic to represent a probability distribution. We can use Mean as an expected value. Remember we are talking about the mean of the variable not the mean of the probability! But not every possible outcome is always equally likely, so we weight the outcomes according to their probability

Mean of a Discrete Random Variable So, instead of summing the options and dividing by the number of options, we sum each option times its probability! So, we use μ instead of 𝑥 since we are talking about all possibilities. So, μx the mean of random variable x μx = 𝒙 𝒊 𝒑 𝒊 (so what is an average Apgar Score??)

Standard Deviation and Variance of Discrete Random Variables If Mean is out measure of center, we will use Standard Deviation as our measure of spread. Again, variance will be the “average” of the squared difference from the mean, but also weighted by its probability. Var(x)= 𝜎 𝑥 2 = ( 𝑥 𝑖 − 𝜇 𝑥 ) 2 𝑝 𝑖 And so Standard Deviation (x) = 𝝈 𝒙 = ( 𝒙 𝒊 − 𝝁 𝒙 ) 𝟐 𝒑 𝒊 !! So standard deviation of Apgar Score is…? Try p. 355 Check understanding

Continuous Random Variables But Wait! What if there are an infinite number of possible outcomes, such as any real number between 0 and 1? Continuous Random Variable x takes all values between to numbers. We use area under a Density Curve to assign probability distribution(area under curve is 1, so total area is sum of all probabilities). Probability of event x is area under the density curve and above (and/or below) the value x that makes up the event. So ALL INDIVIDUAL OUTCOMES HAVE PROBABILITY ZERO!

Assignment Chapter 6, Section 1 pgs 360 – 362 #14,15,17,18,21,23,25