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AG-Ch-6-Notes-WI10 - Chapter 6 Notes Probability...

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1 Chapter 6 Notes AG 1/28/2010 Chapter 6 Notes - Probability Distributions Randomness The numerical values that a variable assumes are the result of some random phenomenon: o Selecting a random sample for a population o Performing a randomized experiment A random variable is a numerical measurement of the outcome of a random phenomenon. o Use letters near the end of the alphabet, such as x , to symbolize Variables in general A particular value of the random variable o Use a capital letter, such as X , to refer to the random variable itself. Example: Flip a coin three times X=number of heads in the 3 flips; defines the random variable x=2; represents a possible value of the random variable Probability Distribution The probability distribution of a random variable specifies its possible values and their probabilities. o Note: It is the randomness of the variable that allows us to specify probabilities for the outcomes A discrete random variable X has separate values (such as 0,1,2,…) as its pos sible outcomes they are countable . o Its probability distribution assigns a probability P (x) to each possible value x : For each x , the probability P (x) falls between 0 and 1 The sum of the probabilities for all the possible x values equals 1 The Mean of a Discrete Probability Distribution The mean of a probability distribution for a discrete random variable is o where the sum is taken over all possible values of x. Expected Value of X o The mean of a probability distribution of a random variable X is also called the expected value of X . The expected value reflects not what we’ll observe in a single observation, but rather that we expect for the average in a long run of observations. It is not unusual for the expected value of a random variable to equal a number that is NOT a possible outcome. The Standard Deviation of a Probability Distribution The standard deviation of a probability distribution , denoted by the parameter, σ , measures its spread. o Larger values of σ correspond to greater spread. o Roughly, σ describes how far the random variable falls, on the average, from the mean of its distribution ) ( x p x
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2 Chapter 6 Notes AG 1/28/2010 Continuous Random Variable A continuous random variable has an infinite number of possible values in an interval. Its probability distribution is specified by a curve. Each interval has probability between 0 and 1. The interval containing all possible values has probability equal to 1.
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