Stats 309 8-1 to 8-2

# Stats 309 8-1 to 8-2 - 8.1 Probability Density Functions A...

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8.1 Probability Density Functions A continuous random variable is one that can assume an uncountable number of values. It is very different from the discrete random variable. We cannot list the possible values because there is an infinite number of them. And because there is an infinite number of possible values, the probability of each individual value is virtually 0. As a result, we can only determine the probability of a range of values. How do we do that? If we arrange the data into a histogram and then superimpose a smooth curve over the histogram, we could then determine a function, f(x), that approximates the curve. This function, f(x), is called a probability density function . The requirements for a probability density function are: Given a pdf f(x) whose range is a≤x≤b: – f(x) ≥ 0 for all x between a and b . the total area under the curve between a and b is 1. Some of the pdf’s we will deal with will go indefinitely in both directions.

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We would normally use integral calculus to then calculate the area under the curve between specific values. However, for the probability distribution functions that we will deal with, this would be very complicated, in most cases. So, we have developed simpler ways to deal with the most common types of probability density functions. The first common probability density function we will work with is the uniform probability density function . This is where the pdf curve, f(x), is a straight line from a to b . See diagram. The pdf is defined as: a b x f - = 1 ) ( You can see from the graph that we have created a rectangle. Now, whatever probability we seek will also create a rectangle. The probability that x is between x 1 and x 2 is written P(x 1 < x < x 2 ). If we add this to our diagram, we see we have created a another rectangle. The probability that we seek corresponds to the area under the ‘curve’ between x 1 and x 2 .
We easily know how to calculate the area of a rectangle: A = bh. The height is f(x) and the base is the difference between x 1 and x 2 . Example 8.1 page 237 The amount of gasoline sold daily at a service station is uniformly distributed with a minimum of 2000 and a maximum of 5000 gallons. – Find the probability that daily sales will fall between 2500 and 3000 gallons. Draw a picture!!!!!!!

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Stats 309 8-1 to 8-2 - 8.1 Probability Density Functions A...

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