Download Continuous Random Variables: Their Probability Density Functions f

Continuous Random Variables:
Their Probability Density Functions f(x), Cumulative Distribution Functions F(x),
and Expected Values E(X)
Recall: A RANDOM VARIABLE assigns a number to some outcome.
If the possible values are at distinct points, then the random variable is DISCRETE.
If the possible values are an interval of values, then the random variable is CONTINUOUS.
Notation: Use capital letters at the end of the alphabet, X, Y, Z, to define the random variable.
The corresponding lower case letters, x, y, z, to represent the actual values.
Example:
Let X = the time, in seconds, it takes a librarian to check out a patron.
Then, x can theoretically take on any value greater than 0 seconds, i.e. x > 0.
To understand how X behaves, we might take a random sample of, say, 100 patrons, and
time how long it takes for the librarian to process their books. Since X is a continuous
random variable, we could create a histogram of the times to get a picture of how X
varies:
50
Percent
40
30
20
10
0
0
10
20
30
40
50
60
Time, in seconds
From this histogram, we might conclude that it takes most patrons 20 seconds or less to
have their books processed. Remember that this histogram is based only on a sample of
100 patrons. What would happen if it were possible to study the whole population of
patrons? We would have so much data that we could imagine making the rectangles
really, really narrow, so that rather than drawing rectangles, we could represent the data
with a curve. This curve is called a CONTINUOUS PROBABILITY DENSITY FUNCTION.
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Then, the curve for X, the time in seconds, might look something like:
0.05
f(x)
0.04
0.03
0.02
0.01
0.00
0
10
20
30
40
50
60
x, time in seconds
in which again we see that it takes the majority of all patrons fewer than 20 seconds to be
pushed through the door. Yet, it takes some people much longer. We can use a
continuous probability density function to calculate the probability that X takes on a
certain range of values, such as:
What is the probability that a randomly selected librarian takes fewer than 15
seconds to process a patron? That is, what is P(X < 15)?
What is the probability that a randomly selected librarian takes more than 40
seconds to process a patron? That is, what is P(X > 40)?
What is the probability that a randomly selected librarian takes between 15 and 25
seconds to process a patron? That is, what is P(15 < X < 25)?
Probability Density Functions f(x)
The notation for a continuous probability density function is the same as that for a discrete
probability density function, namely, f(x). We typically describe a continuous probability density
function by the formula for its curve, but use a cumulative distribution table to calculate
probabilities. First, let’s discuss the characteristics of continuous probability density functions.
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Characteristics:
1. f(x) 0 for all possible values of x. That is, probabilities are (of course) nonnegative.
2. We calculate probabilities geometrically by equating the area under the curve with
probabilities. That is, P(a X b) is the area bounded by the curve f(x), x = a, x = b,
and the x-axis.
3. The area under the whole curve and above the x-axis must be 1. (Corresponding to
the rule that the probabilities across all x must “sum” to 1).
4. The population average or expected value of X, µ = E(X), is the value of X that
makes the curve “balance.” Think seesaw! (To calculate the actual value of E(X), we
would need to use calculus techniques.)
5. The probability that X takes on a specific value is 0. That is X = a is a line, which has
no area, so P(X = a) is 0.
Example:
Let X = the amount of time required for a nurse to respond to a patient’s call. It is known
that X is UNIFORMLY DISTRIBUTED over a 4-minute interval. That is:
0.30
0.25
f(x)
0.20
0.15
0.10
0.05
0.00
0
1
2
3
4
x, in minutes
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Cumulative Distribution Function F(x)
Similar to the discrete case, the CONTINUOUS CUMULATIVE DISTRIBUTION FUNCTION F(x) is the
probability that X is less than or equal to some value. That is, again:
F(x) = P(X x)
So, F(a) = P(X a) is found by finding the area under the curve up to the value a.
Example (cont'd):
What is the probability that a randomly selected patient’s call takes a nurse less than 2
minutes to respond?
What is the probability that a randomly selected patient’s call takes a nurse more than
1 minute to respond?
What is the probability that a randomly selected patient’s call takes a nurse between 1
and 3 minutes to respond?
What is the probability that a randomly selected patient’s call takes a nurse more than
4 minutes to respond?
Now, in practice, the curves that we typically see in nature do not have such a “neat” shape.
Therefore, we will rarely (if ever?) be able to calculate the probabilities associated with a
continuous random variable geometrically. Instead, we must rely on cumulative distribution
tables, in which the cumulative probabilities have already been computed for us.
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