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Section 7.1 Understanding Random Variables What is a RANDOM variable? By the BOOK: A random variable is a variable whose value is a numerical outcome of a random phenomenon. You will be expected to DEFINE a random variable by clearly describing it based on the given information. Example Scenario You are taking a five question True/False quiz in your AP Statistics course. You are really concerned with how many questions you are able to get correct. Define the random variable, X, that is present in this situation. The random variable X is defined as the number of questions answered correctly. DISCRETE Random Variable By the BOOK: A discrete random variable X has a countable number of possible values. The probability distribution for a discrete random variable will look very similar to the probability models from last chapter. Discrete Random Variable Example Recently a friend of yours created a game. He tells you that you are going to get three cards in exchange for two dollars. Your winnings will be based on the number of spades you get. You will get a dollar for each spade. Define the random variable X and describe the probability distribution. Example Solution X = the number of dollars profited X -2 -1 0 1 P(X) .4135 .4359 .1376 .0129 CONTINUOUS Random Variable By the BOOK: A continuous random variable X takes all values in an interval of numbers. The probability distribution of a continuous random variable will be described by a density curve. Probabilities will be calculated based on area underneath the curve. Any density curve (uniform, normal, etc.) can be used to describe a continuous random variable. Continuous Random Variable Example In high school there is always focus on GPA. At Walton High School students tend to do better than they do nationally. It has been established that Walton GPAs are normally distributed with a mean of 3.1 and a standard deviation of 0.47. Define the random variable, X, and describe the probability distribution. Example Solution X = the amount of grade points obtained per class on average The probability distribution of X can be defined by N(3.1, 0.47). Other Important Information Recall, the probability that a continuous random variable takes one specific value will be zero (0) because there is no area under the curve for one point. To get a clear picture of a probability distribution you can look at a probability histogram (for discrete) or the density curve (for continuous). Defining Random Variables As a general guideline, when you are defining a DISCRETE random variable, start with the phrase “the number of . . .” When you are defining a CONTINUOUS random variable, start with the phrase “the amount of . . .” This is often harder to verbalize, but by trying to use this phrase it will hopefully help clarify what your variable represents Review Problems A. Assume that the lengths of salmon are normally distributed with a mean of 10.5 inches and a standard deviation of 2.3 inches. 1. What is the probability of catching a salmon that is more than a foot long? At least a foot long? P(X >12)=P(X≥12)=normalcdf(12,E99,10.5,2.3)=0.2571 B. Consider choosing a number,X, between 0 and 5 where the distribution is uniform. 2. P(X=2) = 0 3. P(X>2.3) = 0.54 4. P(X<1.4 or X>4.8) = 0.32