Probability Density Functions
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- Buck Benson
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2 Recall that a random variable X is continuous if
3 Recall that a random variable X is continuous if 1). possible values of X comprise either a single interval on the number line (for some A < B, any number x between A and B is a possible value) or a union of disjoint intervals;
4 Recall that a random variable X is continuous if 1). possible values of X comprise either a single interval on the number line (for some A < B, any number x between A and B is a possible value) or a union of disjoint intervals; 2). P(X = c) = 0 for any number c that is a possible value of X.
5 Recall that a random variable X is continuous if 1). possible values of X comprise either a single interval on the number line (for some A < B, any number x between A and B is a possible value) or a union of disjoint intervals; 2). P(X = c) = 0 for any number c that is a possible value of X. Examples: 1. X = the temperature in one day. X can be any value between L and H, where L represents the lowest temperature and H represents the highest temperature.
6 Recall that a random variable X is continuous if 1). possible values of X comprise either a single interval on the number line (for some A < B, any number x between A and B is a possible value) or a union of disjoint intervals; 2). P(X = c) = 0 for any number c that is a possible value of X. Examples: 1. X = the temperature in one day. X can be any value between L and H, where L represents the lowest temperature and H represents the highest temperature. 2. Example 4.3: X = the amount of time a randomly selected customer spends waiting for a haircut before his/her haircut commences.
7 Recall that a random variable X is continuous if 1). possible values of X comprise either a single interval on the number line (for some A < B, any number x between A and B is a possible value) or a union of disjoint intervals; 2). P(X = c) = 0 for any number c that is a possible value of X. Examples: 1. X = the temperature in one day. X can be any value between L and H, where L represents the lowest temperature and H represents the highest temperature. 2. Example 4.3: X = the amount of time a randomly selected customer spends waiting for a haircut before his/her haircut commences. Is X really a continuous rv?
8 Recall that a random variable X is continuous if 1). possible values of X comprise either a single interval on the number line (for some A < B, any number x between A and B is a possible value) or a union of disjoint intervals; 2). P(X = c) = 0 for any number c that is a possible value of X. Examples: 1. X = the temperature in one day. X can be any value between L and H, where L represents the lowest temperature and H represents the highest temperature. 2. Example 4.3: X = the amount of time a randomly selected customer spends waiting for a haircut before his/her haircut commences. Is X really a continuous rv? No.
9 Recall that a random variable X is continuous if 1). possible values of X comprise either a single interval on the number line (for some A < B, any number x between A and B is a possible value) or a union of disjoint intervals; 2). P(X = c) = 0 for any number c that is a possible value of X. Examples: 1. X = the temperature in one day. X can be any value between L and H, where L represents the lowest temperature and H represents the highest temperature. 2. Example 4.3: X = the amount of time a randomly selected customer spends waiting for a haircut before his/her haircut commences. Is X really a continuous rv? No. The point is that there are customers lucky enough to have no wait whatsoever before climbing into the barber s chair, which means P(X = 0) > 0. Only conditioned on no chairs being empty, the waiting time will be continuous.
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11 Let s consider the temperature example again. We want to know the probability that the temperature is in any given interval. For example, what s the probability for the temperature between 70 and 80?
12 Let s consider the temperature example again. We want to know the probability that the temperature is in any given interval. For example, what s the probability for the temperature between 70 and 80? Ultimately, we want to know the probability distribution for X.
13 Let s consider the temperature example again. We want to know the probability that the temperature is in any given interval. For example, what s the probability for the temperature between 70 and 80? Ultimately, we want to know the probability distribution for X. One way to do that is to record the temperature from time to time and then plot the histogram.
14 Let s consider the temperature example again. We want to know the probability that the temperature is in any given interval. For example, what s the probability for the temperature between 70 and 80? Ultimately, we want to know the probability distribution for X. One way to do that is to record the temperature from time to time and then plot the histogram. However, when you plot the histogram, it s up to you to choose the bin size.
15 Let s consider the temperature example again. We want to know the probability that the temperature is in any given interval. For example, what s the probability for the temperature between 70 and 80? Ultimately, we want to know the probability distribution for X. One way to do that is to record the temperature from time to time and then plot the histogram. However, when you plot the histogram, it s up to you to choose the bin size. But if we make the bin size finer and finer (meanwhile we need more and more data), the histogram will become a smooth curve which will represent the probability distribution for X.
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19 Definition 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 with a b, P(a X b) = b a f (x)dx That is, the probability that X takes on a value in the interval [a, b] is the area above this interval and under the graph of the density function. The graph of f (x) is often referred to as the density curve.
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21 Figure: P(60 X 70)
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23 Remark: For f (x) to be a legitimate pdf, it must satisfy the following two conditions:
24 Remark: For f (x) to be a legitimate pdf, it must satisfy the following two conditions: 1. f (x) 0 for all x;
25 Remark: For f (x) to be a legitimate pdf, it must satisfy the following two conditions: 1. f (x) 0 for all x; 2. f (x)dx = area under the entire graph of f (x) = 1.
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27 Example: A clock stops at random at any time during the day. Let X be the time (hours plus fractions of hours) at which the clock stops. The pdf for X is known as { 1 f (x) = 24 0 x 24 0 otherwise
28 Example: A clock stops at random at any time during the day. Let X be the time (hours plus fractions of hours) at which the clock stops. The pdf for X is known as { 1 f (x) = 24 0 x 24 0 otherwise The density curve for X is showed below:
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30 Example: (continued) A clock stops at random at any time during the day. Let X be the time (hours plus fractions of hours) at which the clock stops. The pdf for X is known as { 1 f (x) = 24 0 x 24 0 otherwise
31 Example: (continued) A clock stops at random at any time during the day. Let X be the time (hours plus fractions of hours) at which the clock stops. The pdf for X is known as { 1 f (x) = 24 0 x 24 0 otherwise If we want to know the probability that the clock will stop between 2:00pm and 2:45pm, then P(14 X 14.75) = dx = = 1 32
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33 Definition A continuous rv X is said to have a uniform distribution on the interval [A, B], if the pdf of X is f (x; A, B) = { 1 A B A x B 0 otherwise
34 Definition A continuous rv X is said to have a uniform distribution on the interval [A, B], if the pdf of X is f (x; A, B) = { 1 A B A x B 0 otherwise The graph of any uniform pdf looks like the graph in the previous example:
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36 Comparisons between continuous rv and discrete rv:
37 Comparisons between continuous rv and discrete rv: For discrete rv Y, each possible value is assigned positive probability; For continuous rv X, the probability for any single possible value is 0!
38 Comparisons between continuous rv and discrete rv: For discrete rv Y, each possible value is assigned positive probability; For continuous rv X, the probability for any single possible value is 0! P(X = c) = c c f (x)dx = lim ɛ 0 c+ɛ c ɛ f (x)dx = 0
39 Comparisons between continuous rv and discrete rv: For discrete rv Y, each possible value is assigned positive probability; For continuous rv X, the probability for any single possible value is 0! P(X = c) = c c f (x)dx = lim ɛ 0 c+ɛ c ɛ f (x)dx = 0 Since P(X = c) = 0 for continuous rv X and P(Y = c ) > 0,
40 Comparisons between continuous rv and discrete rv: For discrete rv Y, each possible value is assigned positive probability; For continuous rv X, the probability for any single possible value is 0! P(X = c) = c c f (x)dx = lim ɛ 0 c+ɛ c ɛ f (x)dx = 0 Since P(X = c) = 0 for continuous rv X and P(Y = c ) > 0, we have P(a X b) = P(a < X < b) = P(a < X b) = P(a X < b) while P(a Y b ), P(a < Y < b ), P(a < Y b ) and P(a Y < b ) are different.
41 Cumulative Distribution Functions
42 Cumulative Distribution Functions Definition The cumulative distribution function F (x) for a continuous rv X is defined for every number x by F (x) = P(X x) = x f (y)dy For each x, F (x) is the area under the density curve to the left of x.
43 Cumulative Distribution Functions Definition The cumulative distribution function F (x) for a continuous rv X is defined for every number x by F (x) = P(X x) = x f (y)dy For each x, F (x) is the area under the density curve to the left of x.
44 Cumulative Distribution Functions
45 Cumulative Distribution Functions Example 4.6 Let X, the thickness of a certain metal sheet, have a uniform distribution on [A, B]. The pdf for X is f (x) = { 1 B A A x B 0 otherwise Then the cdf for X is calculated as following:
46 Cumulative Distribution Functions Example 4.6 Let X, the thickness of a certain metal sheet, have a uniform distribution on [A, B]. The pdf for X is f (x) = { 1 B A A x B 0 otherwise Then the cdf for X is calculated as following: For x < A, F (x) = 0;
47 Cumulative Distribution Functions Example 4.6 Let X, the thickness of a certain metal sheet, have a uniform distribution on [A, B]. The pdf for X is f (x) = { 1 B A A x B 0 otherwise Then the cdf for X is calculated as following: For x < A, F (x) = 0; for A x < B, we have F (x) = x f (y)dy = x A 1 B A dy = 1 B A y y=x y=a = x A B A ;
48 Cumulative Distribution Functions Example 4.6 Let X, the thickness of a certain metal sheet, have a uniform distribution on [A, B]. The pdf for X is f (x) = { 1 B A A x B 0 otherwise Then the cdf for X is calculated as following: For x < A, F (x) = 0; for A x < B, we have F (x) = x f (y)dy = for x B, F (x) = 1. x A 1 B A dy = 1 B A y y=x y=a = x A B A ;
49 Cumulative Distribution Functions Example 4.6 Let X, the thickness of a certain metal sheet, have a uniform distribution on [A, B]. The pdf for X is f (x) = { 1 B A A x B 0 otherwise Then the cdf for X is calculated as following: For x < A, F (x) = 0; for A x < B, we have F (x) = x f (y)dy = x A 1 B A dy = 1 B A y y=x y=a = x A B A ; for x B, F (x) = 1. Therefore the entire cdf for X is 0 x < A F (x) = x A B A A x < B 1 x B
50 Cumulative Distribution Functions
51 Cumulative Distribution Functions Proposition Let X be a continuous rv with pdf f (x) and cdf F (x). Then for any number a, P(X > a) = 1 F (a) and for any two numbers a and b with a < b, P(a X b) = F (b) F (a).
52 Cumulative Distribution Functions Proposition Let X be a continuous rv with pdf f (x) and cdf F (x). Then for any number a, P(X > a) = 1 F (a) and for any two numbers a and b with a < b, P(a X b) = F (b) F (a).
53 Cumulative Distribution Functions
54 Cumulative Distribution Functions Example (Problem 15) Let X denote the amount of space occupied by an article placed in a 1-ft 3 packing container. The pdf of X is { 90x 8 (1 x) 0 < x < 1 f (x) = 0 otherwise Then what is P(X 0.5) and P(0.25 < X 0.5)?
55 Cumulative Distribution Functions
56 Cumulative Distribution Functions Proposition If X is a continuous rv with pdf f (x) and cdf F (x), then at every x at which the derivative F (x) exists, F (x) = f (x).
57 Cumulative Distribution Functions Proposition If X is a continuous rv with pdf f (x) and cdf F (x), then at every x at which the derivative F (x) exists, F (x) = f (x). e.g. for the previous example, we know the cdf for X is 0 x 0 F (x) = 10x 9 9x 10 0 < x < 1 1 x 1 Then the derivative of F (x) exists on (, ) and we get F (x) = 90x 8 90x 9 for 0 < x < 1 and F (x) = 0 for < x 0 and 1 x <, which is just the pdf of X.
58 Cumulative Distribution Functions
59 Cumulative Distribution Functions Definition The expected value or mean valued of a continuous rv X with pdf f (x) is µ X = E(X ) = x f (x)dx
60 Cumulative Distribution Functions Definition The expected value or mean valued of a continuous rv X with pdf f (x) is µ X = E(X ) = x f (x)dx Definition The variance of a continuous random variable X with pdf f (x) and mean value µ is σx 2 = V (X ) = (x µ) 2 f (x)dx = E[(X µ) 2 ] The standard deviation (SD) of X is σ X = V (X ).
61 Cumulative Distribution Functions
62 Cumulative Distribution Functions Proposition V (X ) = E(X 2 ) [E(X )] 2
63 Cumulative Distribution Functions Proposition V (X ) = E(X 2 ) [E(X )] 2 e.g. for the previous example, the pdf of X is given as { 90x 8 (1 x) 0 < x < 1 f (x) = 0 otherwise
64 Cumulative Distribution Functions Proposition V (X ) = E(X 2 ) [E(X )] 2 e.g. for the previous example, the pdf of X is given as { 90x 8 (1 x) 0 < x < 1 f (x) = 0 otherwise Then the expected value of X is E(X ) = = x f (x)dx = 1 0 x 90x 8 (1 x)dx (x 9 x 10 )dx = 90( 1 10 x x 11 ) x=1 x=0= 9 11
65 Cumulative Distribution Functions
66 Cumulative Distribution Functions Example continued: the pdf for X is { 90x 8 (1 x) 0 < x < 1 f (x) = 0 otherwise
67 Cumulative Distribution Functions Example continued: the pdf for X is { 90x 8 (1 x) 0 < x < 1 f (x) = 0 otherwise The variance of X is V (X ) = E(X 2 ) [E(X )] 2 = = 1 = x 2 90x 8 (1 x)dx [ 0 (x 10 x 11 )dx ( 9 11 )2 = 90( 1 11 x x 12 ) x=1 x=0 ( 9 11 )2 = = x 2 f (x)dx [ 0 x 90x 8 (1 x)dx] 2 x f (x)dx] 2
68 Cumulative Distribution Functions
69 Cumulative Distribution Functions Definition Let p be a number between 0 and 1. The (100p)th percentile of the distribution of a continuous rv X, denoted by η(p), is defined by p = F (η(p)) = η(p) f (y)dy
70 Cumulative Distribution Functions Definition Let p be a number between 0 and 1. The (100p)th percentile of the distribution of a continuous rv X, denoted by η(p), is defined by p = F (η(p)) = η(p) f (y)dy In words, the (100p)th percentile η(p) is the X value such that there are 100p% X values below η(p).
71 Cumulative Distribution Functions Definition Let p be a number between 0 and 1. The (100p)th percentile of the distribution of a continuous rv X, denoted by η(p), is defined by p = F (η(p)) = η(p) f (y)dy In words, the (100p)th percentile η(p) is the X value such that there are 100p% X values below η(p). Graphically, η(p) is the value on the measurement axis such that 100p% of the area under the graph of f (x) lies to the left of η(p) and 100(1 p)% lies to the right.
72 Cumulative Distribution Functions
73 Cumulative Distribution Functions
74 Cumulative Distribution Functions
75 Cumulative Distribution Functions Definition The median of a continuous distribution, denoted by µ, is the 50th percentile, so µ satisfies 0.5 = F ( µ). That is, half the area under the density curve is to the left of µ and half is to the right of µ.
76 Cumulative Distribution Functions Definition The median of a continuous distribution, denoted by µ, is the 50th percentile, so µ satisfies 0.5 = F ( µ). That is, half the area under the density curve is to the left of µ and half is to the right of µ. e.g. for the continuous rv X with cdf 0 x 0 F (x) = 10x 9 9x 10 0 < x < 1 1 x 1 the 100pth percentile is calculated as following: p = F (η(p)) = 10η(p) 9 9η(p) 10
77 Cumulative Distribution Functions Definition The median of a continuous distribution, denoted by µ, is the 50th percentile, so µ satisfies 0.5 = F ( µ). That is, half the area under the density curve is to the left of µ and half is to the right of µ. e.g. for the continuous rv X with cdf 0 x 0 F (x) = 10x 9 9x 10 0 < x < 1 1 x 1 the 100pth percentile is calculated as following: p = F (η(p)) = 10η(p) 9 9η(p) 10 Therefore, the 75th percentile is η(.75) and the median is η(.5)
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