# 37 - Chapter 6 Probability Distributions Section 6.1 How...

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1 Chapter 6: Probability Distributions Section 6.1: How Can We Summarize Possible Outcomes and Their Probabilities?

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2 Learning Objectives 1. Random variable 2. Probability distributions for discrete random variables 3. Mean of a probability distribution 4. Summarizing the spread of a probability distribution 5. Probability distribution for continuous random variables
3 Learning Objective 1: Randomness The numerical values that a variable assumes are the result of some random phenomenon: Selecting a random sample for a population or Performing a randomized experiment

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4 Learning Objective 1 : Random Variable A random variable is a numerical measurement of the outcome of a random phenomenon.
5 Learning Objective 1: Random Variable Use letters near the end of the alphabet, such as x , to symbolize Variables A particular value of the random variable 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

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6 Learning Objective 2: Probability Distribution The probability distribution of a random variable specifies its possible values and their probabilities. Note: It is the randomness of the variable that allows us to specify probabilities for the outcomes
7 Learning Objective 2 : Probability Distribution of a Discrete Random Variable A discrete random variable X has separate values (such as 0,1,2,…) as its possible outcomes 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

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8 Learning Objective 2: Example What is the estimated probability of at least three home runs? P(3)+P(4)+P(5)=0.13+0.03+0.01=0.17
9 Learning Objective 3 : The Mean of a Discrete Probability Distribution The mean of a probability distribution for a discrete random variable is where the sum is taken over all possible values of x. = ) ( x p x μ The mean of a probability distribution is denoted by the parameter, µ. The mean is a weighted average; values of x that are more likely receive greater weight P(x)

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10 Learning Objective 3 : Expected Value of X 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.
Learning Objective 3: Example Find the mean of this probability distribution. The mean:

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37 - Chapter 6 Probability Distributions Section 6.1 How...

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