1. Probability Densities
Chapter 5
Probability Densities
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

2. Continuous Random Variables
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

3. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Random Variable
For a given sample space S of some experiment, a random variable is any rule that associates a number with each outcome in S .
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

4. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Types of Random Variables
A discrete random variable is an rv whose possible values either constitute a finite set or else can listed in an infinite sequence. A random variable is continuous if its set of possible values consists of an entire interval on a number line.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

5. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Random Variables
Represents a possible numerical value from a random event
Random
Variables
Discrete
Random Variable
Continuous
Random Variable
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

6. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Continuous Random Variables
A random variable X is continuous if its set of possible values is an entire interval of numbers (If A < B, then any number x between A and B is possible).
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

7. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Continuous Probability Distribution
Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f (x) such that for any two numbers a and b,
The graph of f is the density curve.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

8. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Probability Density Function
For f (x) to be a pdf
f (x) > 0 for all values of x.
The area of the region between the graph of f and the x – axis is equal to 1.
Area = 1
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

9. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Probability Density Function
is given by the area of the shaded region. a b
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

10. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Probability for a Continuous rv
If X is a continuous rv, then for any number c, P(x = c) = 0. For any two numbers a and b with a < b,
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

11. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Cumulative Distribution Function
The cumulative distribution function, F(x) for a continuous rv X is defined for every number x by
For each x, F(x) is the area under the density curve to the left of x.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

12. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Using F(x) to Compute Probabilities
Let X be a continuous rv with pdf f(x) and cdf F(x). Then for any number a,
and for any numbers a and b with a < b,
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

13. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Obtaining f(x) from F(x)
If X is a continuous rv with pdf f(x) and cdf F(x), then at every number x for which the derivative exists,
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

14. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Percentiles
Let p be a number between 0 and 1. The (100p)th percentile of the distribution of a continuous rv X denoted by , is defined by
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

15. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Median
The median of a continuous distribution, denoted by , is the 50th percentile. So satisfies That is, half the area under the density curve is to the left of
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

16. of Continuous Random Variables
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

17. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Expected Value
The expected or mean value of a continuous rv X with pdf f (x) is
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

18. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Expected Value of h(X)
If X is a continuous rv with pdf f(x) and h(x) is any function of X, then
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

19. Variance and Standard Deviation
The variance of continuous rv X with pdf f(x) and mean is
The standard deviation is
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

20. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Short-cut Formula for Variance
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

21. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Covariance between two discrete random variables:
σxy =  [x – E(x)] [y – E(y)] f(xy)
where:
xi = possible values of the x discrete random variable
yj = possible values of the y discrete random variable
P(xi ,yj) = joint probability of the values of xi and yj occurring
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

22. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Let X and Y be random variables, The Covariance of X and Y is
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

23. Correlation Coefficient
The Correlation Coefficient shows the strength of the linear association between two variables
where:
ρ = correlation coefficient (“rho”)
σxy = covariance between x and y
σx = standard deviation of variable x
σy = standard deviation of variable y
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

24. Interpreting the Correlation Coefficient
The Correlation Coefficient always falls between -1 and +1
 = x and y are not linearly related.
The farther  is from zero, the stronger the linear relationship:
 = x and y have a perfect positive linear relationship
 = x and y have a perfect negative linear relationship
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

25. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
4.3
Mean and Variance of Linear Combinations of Random Variables
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

26. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Expected Value of Linear Combinations
This leads to the following:
For any constant a,
2. For any constant b,
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

27. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Expected Value of
Also
Finally
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

28. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
If X and Y are independent, then
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

29. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Expected value of the sum of two discrete random variables:
E(x + y) = E(x) + E(y)
=  x P(x) +  y P(y)
(The expected value of the sum of two random variables is the sum of the two expected values)
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

30. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Rules of Variance
and
This leads to the following:
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

31. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
If X and Y are random variables, then
If X and Y are Independent random variables, then
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

32. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
If X and Y are Independent random variables, then
If X1, X2, … Xn are Independent random variables, then
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

33. Special Continuous Probability Densites
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

34. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Uniform Distribution
A continuous rv X is said to have a uniform distribution on the interval [A, B] if the pdf of X is
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

35. The Normal Distribution
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

36. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Normal Distributions
A continuous rv X is said to have a normal distribution with parameters
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

37. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Standard Normal Distributions
The normal distribution with parameter values is called a standard normal distribution. The random variable is denoted by Z. The pdf is
The cdf is
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

38. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Standard Normal Cumulative Areas
Standard normal curve z
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

39. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Standard Normal Distribution
Let Z be the standard normal variable. Find (from table) a.
Area to the left of =
b. P(Z > 1.32)
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

40. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Find the area to the left of 1.78 then subtract the area to the left of –2.1.
= – =
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

41. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Notation
will denote the value on the measurement axis for which the area under the z curve lies to the right of
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

42. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Let Z be the standard normal variable. Find z if a. P(Z < z) =
Look at the table and find an entry = then read back to find
z = 1.46.
b. P(–z < Z < z) =
P(z < Z < –z ) = 2P(0 < Z < z)
= 2[P(z < Z ) – ½]
= 2P(z < Z ) – 1 =
P(z < Z ) =
z = 1.32
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

43. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Nonstandard Normal Distributions
If X has a normal distribution with mean and standard deviation , then
has a standard normal distribution.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

44. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Normal Curve
Approximate percentage of area within given standard deviations (empirical rule).
99.7%
95%
68%
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

45. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Let X be a normal random variable with =
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

46. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. A particular rash shown up at an elementary school. It has been determined that the length of time that the rash will last is normally distributed with
Find the probability that for a student selected at random, the rash will last for between 3.75 and 9 days.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

47. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
= – =
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

48. Percentiles of an Arbitrary Normal Distribution
(100p)th percentile for normal
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

49. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Normal Approximation to the Binomial Distribution
Let X be a binomial rv based on n trials, each with probability of success p. If the binomial probability histogram is not too skewed, X may be approximated by a normal distribution with
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

50. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. At a particular small college the pass rate of Intermediate Algebra is 72%. If 500 students enroll in a semester determine the probability that at least 375 students pass. =
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

51. The Gamma Distribution
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

52. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Gamma Function
the gamma function
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

53. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Gamma Distribution
A continuous rv X has a gamma distribution if the pdf is
where the parameters satisfy
The standard gamma distribution has
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

54. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Mean and Variance
The mean and variance of a random variable X having the gamma distribution
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

55. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Exponential Distribution
A continuous rv X has an exponential distribution with parameter if the pdf is
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

56. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Mean and Variance
The mean and variance of a random variable X having the exponential distribution
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

57. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Probabilities from the Gamma Distribution
Let X have a exponential distribution
Then the cdf of X is given by
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

58. Applications of the Exponential Distribution
Suppose that the number of events occurring in any time interval of length t has a Poisson distribution with parameter and that the numbers of occurrences in nonoverlapping intervals are independent of one another. Then the distribution of elapsed time between the occurrences of two successive events is exponential with parameter
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

59. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Chi-Squared Distribution
Let v be a positive integer. Then a random variable X is said to have a chi-squared distribution with parameter v if the pdf of X is the gamma density with
The pdf is
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

60. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Chi-Squared Distribution
The parameter v is called the number of degrees of freedom (df) of X. The symbol is often used in place of “chi-squared.”
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

61. Other Continuous Distributions
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

62. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Weibull Distribution
A continuous rv X has a Weibull distribution if the pdf is
where the parameters satisfy
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

63. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Mean and Variance
The mean and variance of a random variable X having the Weibull distribution are
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

64. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Lognormal Distribution
A nonnegative rv X has a lognormal distribution if the rv Y = ln(X) has a normal distribution the resulting pdf has parameters
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

65. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Mean and Variance
The mean and variance of a variable X having the lognormal distribution are
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

66. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Beta Distribution
A rv X is said to have a beta distribution with parameters A, B,
if the pdf of X is
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

67. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Mean and Variance
The mean and variance of a variable X having the beta distribution are
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.