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**Probability Distributions**

Continuous Random Variables

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**Continuous Random Variables**

A random variable was a numerical value associated with the outcome of an experiment. Finite discrete random variables were ones in which the values were countable whole numbered values A continuous random variable is a random variable that can assume any value in some interval of numbers, and are thus NOT countable. Examples: The time that a train arrives at a specified stop The lifetime of a transistor A randomly selected number between 0 and 1 Let R be a future value of a weekly ratio of closing prices for IBM stock Let W be the exact weight of a randomly selected student

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**Continuous Random Variables**

A random variable is said to be continuous if there is a function fX(x) with the following properties: Domain: all real numbers Range: fX(x)≥0 The area under the entire curve is 1 Such a function fX(x) is called the probability density function (abbreviated p.d.f.) The fact that the total area under the curve fX(x) is 1 for all X values of the random variable tells us that all probabilities are expressed in terms of the area under the curve of this function. Example: If X are values on the interval from [a,b], then the P(a≤X≤b) = area under the graph of fX(x) over the interval [a,b] A a b fX

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**Continuous Random Variables**

Because all probabilities for a continuous random variable are described in terms of the area under the p.d.f. function, the P(X=x) = 0. Why: the area of the p.d.f. for a single value is zero because the width of the interval is zero! That is, for any continuous random variable, X, P(X = a) = 0 for every number a. This DOES NOT imply that X cannot take on the value a, it simply means that the probability of that event is 0.

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**Continuous Random Variables**

Rather than considering the probability of X taking on a given single value, we look for the probability that X assumes a value in an interval. Suppose that a and b are real numbers with a < b. Recall that X a is the event that X assumes a value in the interval(, a]. Likewise, a < X b and b < X are the events that X assumes values in (a, b] and (b, ), respectively. These three events are mutually exclusive and at least one of them must happen. Thus, P(X a) + P(a < X b) + P(b < X) = 1. Since we are interested in the probability that X takes a value in an interval, we will solve for P(a < X b).

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**Continuous Random Variables**

Because X is a continuous random variable, P(X = a) = 0 and P(X = b) = 0. Thus, it makes no difference whether or not we include the end points in an interval.

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**The cumulative distribution function**

The same probability information is often given in a different form, called the cumulative distribution function, (c.d.f), FX(x) FX(x)=P(Xx) 0 FX(x) 1, for all x Domain is all real numbers

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Example The p.d.f. of T, the weekly CPU time (in hours) used by an accounting firm, is given below.

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Example (cont) The graph of the p.d.f. is given below:

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Example (cont) is equal to the area between the graph of and the t-axis over the interval.

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Another Example The c.d.f. of T (for the previous example) is given below. Find

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The graph of the c.d.f.

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Solution

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**Expected Value—Continuous Random Variable**

For discrete finite random variables, the expected value was determined by taking each value of the random variable and multiplying it by the corresponding probability as stated by:

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**Expected Value (cont) X=x fX(x) x fX(x) fX(0) 0 fX(0) 1 fX(1) 1 fX(1)**

For a continuous random variable, the process is more tricky. For example, suppose we wanted to know the expected value of X which is defined on the interval [0,2]. Unfortunately, the P(X=a) for any number is 0 So when we try to compute the expected value we’re adding up a whole bunch of zeros X=x fX(x) x fX(x) fX(0) 0 fX(0) 1 fX(1) 1 fX(1) 2 fX(2) 2 fX(2)

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Expected Value (cont) We need some other way of looking for the expected value for a continuous random variable. Unfortunately, this requires calculus which you will learn more about in 115b Basically we need to find the area under the entire curve for the function x fX(x) We can’t do this without knowing some calculus so we’ll give a more geometric interpretation of the E(X) for a continuous random variable.

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**Expected Value—Geometric Interpretation**

We can use the probability density function to give a geometric interpretation for the mean of a continuous random variable, X. Suppose that we draw the p.d.f. on a thin sheet of metal and cut out the region between the graph of f(X) and the x-axis. If we place a knife edge under a line through on the x-axis and perpendicular to that axis, then the metal sheet will balance on that edge. X fX

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**A more concrete example**

Consider the p.d.f. shown below for a continuous random variable X

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**Expected Value—Geometric Interpretation**

Note: The mean does not occur at the highest point of the graph! x-coordinate of highest point is the mode. In the previous example, this was at x = 1 Note: The mean does not divide the area in half! x-coordinate that does this is the median. In previous example, the median is approximately

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Special Distribution A continuous uniform random variable is a random variable defined on an interval such that every subinterval of having the same length has the same probability. If X is a continuous uniform random variable on the interval , then

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**Example—Uniform Distribution**

A bus arrives at a bus stop every 10 minutes. Let W be the waiting time (in minutes) until the next bus. The p.d.f. and c.d.f. of W are given below.

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**Uniform Distribution: p.d.f. & c.d.f.**

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**Questions: Find Find E(X)**

Notice that The expected value of W is given by: .

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Special Distribution An exponential random variable may be used to model the length of time between consecutive occurrences of some event in a fixed unit of space or time. If X is an exponential random variable with parameter, then

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Example On average, three customers per hour use the ATM in a local grocery store. Let T be the time (in minutes) between consecutive customers. The p.d.f. and c.d.f. of T are given below.

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**Exponential Distribution: p.d.f. & c.d.f.**

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Questions: Find The expected value cannot be determined from the p.d.f. function without using calculus so we’ll simply tell you that: The expected value of T is given by:

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**Summary of Distributions**

fX(x) x

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**Summary of Distributions**

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**Expected Values Finite Discrete Random Variables:**

Continuous Random Variables: Uniform Distribution: Exponential Distribution:

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**Additional Information**

See Distribution Study Guide under “Worksheets” link on my webpage Make sure you are able to distinguish between the various types of distributions, their expected values, the c.d.f. and p.d.f.

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