# 3 we can express f x in multiple ways 1 table of x

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We can express f ( x ) in multiple ways. 1. Table of x and f ( x ): 2. Graphically: 3. Formula Ex : Suppose x is a discrete random variable that takes on the values 1, 2, 3, and 4, each with probability 1 4 . This is an example of a discrete uniform probability function. f ( x ) = 1 4 4
Ex : The probability distribution for the random variable x is below. x f ( x ) 20 0.20 25 0.15 30 0.25 35 0.40 a. Is this probability distribution valid? b. What is the probability that x = 30? c. What is the probability that x is less than or equal to 25? d. What is the probability that x is greater than 30? 5

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Expected Value or Mean We can compute a measure of central location for a discrete random variable. This measure is called the expected value or mean of the discrete random variable. E ( x ) = μ = X xf ( x ) Ex : A volunteer ambulance service handles 0 to 5 service calls on any given day. The probability distribution for the number of service calls is below. What is the expected number of service calls? Number Probability 0 0.10 1 0.15 2 0.30 3 0.20 4 0.15 5 0.10 6
Variance We can also measure the variability for a discrete random variable. This measure is called the variance of the discrete random variable. V ar ( x ) = σ 2 = X ( x - μ ) 2 f ( x ) The standard deviation is the square root of the variance. Ex : What is the variance in the number of service calls? What is the standard deviation? 7

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Ex : The American Housing Survey reported the following data on the number of bedrooms in renter-occupied houses in central cities. Bedrooms Number 0 547 1 5012 2 6100 3 2644 4 557 a. Define a random variable x = number of bedrooms in renter-occupied houses. Develop a probability distribution for the random variable. b. Compute the expected value for the number of bedrooms in renter- occupied houses. c. Compute the variance and standard deviation for the number of bed- rooms in renter-occupied houses. 8
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