Continuous Probability Distributions

Continuous Random VariableA random variable whose space (set of possible

values) is an entire interval of numbers

Probability Density Function (or pdf)The pdf, denoted f( x ), describes the distribution of

probability across the set of possible values.The probability that the random variable X takes on a

value between a and b is equal to the area under f( x ) between a and b.

Probability Density Function

The pdf of a continuous random variable X has the following properties:

1. f( x ) ≥ 0, for every x S

2. -∞∞

f( x ) dx = 1

3. P( x A ) = A f( x ) dx

Probability Density Function

So we have that: P( a < x < b ) = a

b f( x ) dx

P( X = a ) = aa

f( x ) dx = 0

P( a ≤ x ≤ b ) = P( a < x < b )

Histogram

A “connected” bar plot with bar height proportional to the frequency of the associated class. Can be very useful for estimating a pdf.

Discrete Data Each distinct outcome is marked on horizontal axis. Bar is plotted atop each distinct outcome with bar

height equal to frequency or relative frequency or that outcome.

Histogram

Continuous Data Construct a frequency table

Partition data into classes and tabulate the frequency for each class.

Use frequency table to plot histogramMark class boundaries on horizontal axis and plot

a bar on top of each class with height equal to the frequency or relative frequency of data in that class.

Cumulative Distribution Function

Called the cdf or distribution function

The cdf of a continuous random variable X is given by:

F( x ) = P( X ≤ x ) = -∞x

f( t ) dt

Consider that P( a < X < b) = P( a ≤ x ≤ b ) = F( b ) – F( a ) P( X > a ) = 1 – P( X ≤ a ) = 1 – F( a )

Cumulative Distribution Function

Using the fundamental theorem of calculus, it can be shown that:

'( ) ( ) ( ) ( )xd d

F x F x f t dt f xdx dx

Percentile

Let p be a number between 0 and 1, and let X be a continuous random variable with pdf f( x ) and cdf f( x ).

The 100·pth percentile is the number such that F( pp )=p.

Solve the following equation for pp:

( ) ( )p

pF f x dx p

Mathematical Expectation

The mathematical expectation of a continuous random variable X with pdf f( x ) is:

m = E[ X ] = -∞∞

x f( x ) dx

E[ X ] is also called the mean or expected value of X

Variance

The variance of a continuous random variable X with pdf f( x ) is:

s2 = E[ ( X – m )2 ]

= -∞∞

( x – m )2 f( x ) dx

= E[ X2 ] – (E[ X ])2

measure of spread in f( x )

Expected Value of a Function

The expected value of the function h( x ) of a continuous random variable X with pdf f( x ) is:

m = E[ h( X ) ] = -∞∞

h( x ) f( x ) dx

Note that E[ h( X ) ] might not exist.

Continuous Uniform Distribution

Uniformly distributes the probability across the sample space.

If X is a continuous random variable on the interval [a,b], then

The pdf of X is:

f( x ) = 1 / ( b – a ), for a ≤ x ≤ b

Continuous Uniform Distribution

If X is a continuous random variable on the interval [a,b], then

The cdf of X is:

F( x ) = ( x – a ) / ( b – a ), for a ≤ x ≤ b

The mean and variance of X are:

m = E[ X ] = ( a + b ) / 2

s2 = ( b – a )2 / 12

Exponential Distribution

Can be used to describe the waiting time between successive events in a Poisson process with mean l

If X is an exponential random variable from a Poisson process with mean l, then

The pdf of X is:

f( x ) = le-lx , for x ≥ 0

Exponential Distribution

If X is an exponential random variable from a Poisson process with mean l, then The cdf of X is:

F( x ) = P ( X ≤ x ) = 1 - e-lx, for x ≥ 0

So P ( X > x ) = 1 – F ( x ) = e-lx

The mean and variance of X are:

m = 1/l s2 = 1/l2

Exponential Distribution

Memoryless Property P( X > k + j | X > k ) = P( X > j )

Percentiles of an Exponential The pth percentile of an exponential random

variable with mean 1/l is:

pp = -1/l ln( 1 – p )

Normal Distribution

Commonly occurring distribution in nature and experimental settings. symmetric, bell-shaped

A normal random variable X with mean m and variance s2

is denoted: X ~ N( m, s2 )

has pdf:2

2

1

2

1)(

x

exf

Standard Normal Distribution

A standard normal random variable is denoted Z and has distribution Z ~ N( 0, 1 ). The pdf and cdf of a standard normal random

variable are denoted f( z ) and F( z ) respectively. Table A.3 contains probability values associated

with F( z ). Note that F( z ) = 1 - F( -z )

Standard Normal Distribution

za Notation

The value of Z that has a probability to its right is denoted za, so P( Z > z a ) = a.

by symmetry, P( Z < -z a ) = P( Z > z a ) = a

Percentile

pp = z1-p

Non-Standard Normal Distribution

Any normal random variable X ~ N( m, s2 ) can be “standardized” into a Z by:

Z = ( x – m ) / s

Percentile

pp = m + z1-p s

Normal Approximation of Discrete Distributions

Many discrete random variables can be approximated by the normal distribution

A continuity correction of ±½ is required when estimating a discrete probability with the normal distribution

Normal Approximation of the Binomial Distribution

Let X be a binomial random variable with sample size n and probability of success p. If the sample size is sufficient ( np ≥ 5 and nq ≥ 5 ), then X can be approximated by a normal distribution with the same mean m=np and variance s2=npq.

X ~ B( n, p ) N( np, npq )

PDF for Graphs:Binomial n=15, Binomial n=50, Binomial p=0.25

Normal Approximation of the Poisson Distribution

Let X be a Poisson random variable with sufficiently large mean l, then X can be approximated by a normal distribution with the same mean m=s2=npq.

X ~ P( l ) N( l, l )

PDF for Graphs: Poisson ( l = 1, 5, 10, 15 )

Empirical Rule

For data that is approximately normal in distribution (bell-shaped), 68% of data values fall within 1 standard

deviation of the mean, 95.4% of data values fall within 2 standard

deviation of the mean, 99.7% of data values fall within 3 standard

deviation of the mean,

x - 3s x - 2s x - s x x + 2s x + 3sx + s

68% within1 standard deviation

34% 34%

95% within 2 standard deviations

99.7% of data are within 3 standard deviations of the mean

0.1% 0.1%

2.4% 2.4%

13.5% 13.5%

The Empirical Rule(applies to bell-shaped distributions)

Identifying Unusual Observations Range Rule of Thumb:

Empirical rule says 95% of observations should fall within 2s of the mean. Observations outside of m±2s are considered unusual.

Probability Approach: The probability of the observed outcome or

more extreme can be useful for identifying unusual observations. For example, if X is the observed outcome:X is unusually high if P(x or more) is less than 0.05X is unusually low if P(x or less) is more than 0.05

Probability Plots

A probability plot can be useful for comparing the distribution of one sample data set to another. If both data sets have the same sample size,

then plot the order statistics from each sample against each other in a scatter plot.

Otherwise, plot order statistics from the smaller sample against corresponding sample percentiles from the other sample.Note: The ith smallest observation is taken to

be the (100*(i-½)/n)th sample percentile.

Measures of Relative Position

Percentile The kth percentile (Pk) separates the bottom k%

of data from the top (100-k)% of data. The location of Pk in the order statistics is:

integeran not is

100 if

100Ceiling

integeran is100 if5.0

100knkn

knkn

L

Interpretation of Probability Plots

Probability plot for comparing 2 sample data sets: A straight line with slope 1 and y-intercept 0

indicates identical sample distributions. A slope greater than 1 indicates that x is less

variable than y. A slope less than 1 indicates that x is more

variable than y. A y-intercept different from 0 indicates that the

two samples have a different mean.

Probability Plots

A probability plot can be useful for comparing the distribution of sample data to a specified probability distribution. Order statistics (or sample percentiles) of the

sample are plotted against the corresponding percentiles of the probability distribution of interest.

Called the “Normal Probability Plot” when sample data is compared to normal percentiles.

Simulating Data from a Continuous Probability Distribution Theorem

Let Y be U(0,1), a continuous uniform random variable on the interval (0,1). Let F( x ) have the properties of a cdf, then X = F -1( Y ) is a continuous random variable with cdf F( x ).

So, random data can be generate from any cdf that has an inverse function by generating random U(0,1) data and transforming it into F( x ) data.