Modeling Random Events-The Normal and Bimodal Models (Chapter 6)Lecture 10 Probability Models and Distributions - Probability models: is a statisticians guess of how data is produced - Probability models are UNIFORM, LINEAR, NORMAL, OTHER- A probability distribution or probability distribution function (PDF) is a table, graph or formula that contain all the outcomes of an experiment and their probabilities Here is an example of a probability model for a coin toss:Outcome Heads Tails Probability 1/21 1/2 represents how statisticians would guess outcomes are produces each outcome has the same chanceDiscrete vs. Continuous - A numeric variable is Discrete is the outcomes can be listed or countedo Examples of discrete numeric variables would be: number of classes taken, or number of siblings- A random variable is continuous if the outcomes cant be listed because they occur over a range with an infinite amount of possible values o Examples of continuous random variables would be: obese is defined as a BMI greater than or equal to 30, or low birth weight is defined as less than 53 ounces An example of continuous numeric variable would be birthweight in ounces. We would count this as continuous because there is a certain amount of range involved. An example of discrete numeric variable would be the number of living children a woman has birthed. We would categorize this as discrete because humans are counted as whole units. Test Your Understanding of Discrete or Continuous Discrete or Continuous?Ask yourself how each one is counted? Is there range? Or whole units used?A)length of your left thumbB)number of presents you get for graduating UCLAC)number of electrical outlets in your homeD)Sodium concentration in your bloodstream Answer Key a)C b)D c)D d)CDiscrete Probability Distributions- The most common way to display probability distribution function (pdf) for discrete data is with a table o The Probability distrivution tavle always has two columns(or rows) The first , x displays all the possible outcome The second, P(x) displays the probabilities for these outcomes here is an example of probability distribution table for a rafflex P(x)95 .o1995 .005-5 .985- The sum of all the probabilities in a distribution table MUST equal 1- The probabilities may or may not be equally likely (above, not equally likely)-examples of probability can also be translated in to graph form  discrete probability can also be an equation (BUT we are not asked to memorize this function) Continuous Data and Probability Distributions Continuous variables are a little more complicated - We cant just list out all of the possible outcomes because they are infinite HOWEVER we can work with ranges of values Example: suppose we want to estimate the probability that a nbaby girl will grow to at least 173 cm in height. We cant list out all the probabilities(173.001, 173.003 ect) so we work with ranges instead of exact values  probabilities are represented as curves this is called a probability density cutce - The area under the curve between the two values of x represents the probability of x being between these two values - The total area under the curve must equal 1- The curve cannot lie below the x-axis This area represents (P (0 <x <2)2 50Finding Probabilities for uniform distributions  the curve shows the probability distribution function for time to wait for a bus that comes ever 12 minutes. Fine the probability that you will wait between 5 and 1ominutes - F t shade the area irs  first shade the area  then find the area of the rectangleP(5 <x<10)=Base X Height =5 X 0.8333 =.41655DON’T PAINIC- In stats 10 you wil either use a computer or a printed tale to find the area under the curve for a continuous variable - In practice statisticians use the computer to approximate these areas Next time we will look at continuous probability distributions !Delay.15.10.8333 .05150105Waiting Time