random variables and probability distributions

Random Variables and Probability Distributions

Modified from a presentation by Carlos J. Rosas-Anderson

Fundamentals of Probability

The probability P that an outcome occurs is:

The sample space is the set of all possible outcomes of an event Example: Visit = {(Capture), (Escape)}

trialsofnumberoutcomesofnumberP

Axioms of Probability1. The sum of all the probabilities of

outcomes within a single sample space equals one:

2. The probability of a complex event equals the sum of the probabilities of the outcomes making up the event:

3. The probability of 2 independent events equals the product of their individual probabilities:

)()()( BPAPBorAP

0.1)(1

n

iiAP

)()()( BPAPBandAP

Probability distributions We use probability

distributions because they fit many types of data in the living world

Ht (cm) 1996

100

80

60

40

20

0

Std. Dev = 14.76

Mean = 35.3

N = 713.00

Ex. Height (cm) of Hypericum cumulicola at Archbold Biological Station

Probability distributions Most people are familiar with the Normal

Distribution, BUT… …many variables relevant to biological

and ecological studies are not normally distributed! For example, many variables are discrete

(presence/absence, # of seeds or offspring, # of prey consumed, etc.)

Because normal distributions apply only to continuous variables, we need other types of distributions to model discrete variables.

Random variable The mathematical rule (or function)

that assigns a given numerical value to each possible outcome of an experiment in the sample space of interest.

2 Types: Discrete random variables Continuous random variables

The Binomial DistributionBernoulli Random Variables

Imagine a simple trial with only two possible outcomes: Success (S) Failure (F)

Examples Toss of a coin (heads or tails) Sex of a newborn (male or female) Survival of an organism in a region (live or

die)

Jacob Bernoulli (1654-1705)

The Binomial DistributionOverview

Suppose that the probability of success is p

What is the probability of failure? q = 1 – p

Examples Toss of a coin (S = head): p = 0.5 q = 0.5 Roll of a die (S = 1): p = 0.1667 q = 0.8333 Fertility of a chicken egg (S = fertile): p = 0.8 q = 0.2


Imagine that a trial is repeated n times Examples:

A coin is tossed 5 times A die is rolled 25 times 50 chicken eggs are examined

ASSUMPTIONS: 1) p is constant from trial to trial2) the trials are statistically independent of each

other


What is the probability of obtaining X successes in n trials?

Example What is the probability of obtaining 2 heads

from a coin that was tossed 5 times?

P(HHTTT) = (1/2)5 = 1/32


But there are more possibilities:

HHTTT HTHTT HTTHT HTTTHTHHTT THTHT THTTH

TTHHT TTHTHTTTHH

P(2 heads) = 10 × 1/32 = 10/32


In general, if n trials result in a series of success and failures,

FFSFFFFSFSFSSFFFFFSF…

Then the probability of X successes in that order is

P(X) = q q p q = pX qn – X


However, if order is not important, then

where is the number of ways to obtain X successes

in n trials, and n! = n (n – 1) (n – 2) … 2 1

n!X!(n – X)!

pX qn – XP(X) =

n!X!(n – X)!


Bin(0.1, 5)

00.20.40.60.8

0 1 2 3 4 5

Bin(0.3, 5)

00.10.20.30.4

0 1 2 3 4 5

Bin(0.5, 5)

00.10.20.30.4

0 1 2 3 4 5Bin(0.7, 5)

00.10.20.30.4

0 1 2 3 4 5

Bin(0.9, 5)

00.20.40.60.8

0 1 2 3 4 5

The Poisson DistributionOverview

When there are a large number of trials but a small probability of success, binomial calculations become impractical Example: Number of

deaths from horse kicks in the French Army in different years

The mean number of successes from n trials is λ = np Example: 64 deaths in 20

years out of thousands of soldiers

Simeon D. Poisson (1781-1840)


If we substitute λ/n for p, and let n approach infinity, the binomial distribution becomes the Poisson distribution:

P(x) = e -λλx

x!


The Poisson distribution is applied when random events are expected to occur in a fixed area or a fixed interval of time

Deviation from a Poisson distribution may indicate some degree of non-randomness in the events under study

See Hurlbert (1990) for some caveats and suggestions for analyzing random spatial distributions using Poisson distributions

The Poisson DistributionExample: Emission of -particles

Rutherford, Geiger, and Bateman (1910) counted the number of -particles emitted by a film of polonium in 2608 successive intervals of one-eighth of a minute What is n? What is p?

Do their data follow a Poisson distribution?

The Poisson DistributionEmission of -particles

No. -particles

Observed

057

1203

2383

3525

4532

5408

6273

7139

845

927

1010

114

120

131

141

Over 14 0

Total2608

Calculation of λ:

λ = No. of particles per interval

= 10097/2608= 3.87

Expected values:2608 P(x) = e -3.87(3.87)x

x!2608


No. -particles

Observed

Expected

057 54

1203 210

2383 407

3525 525

4532 508

5408 394

6273 255

7139 140

845 68

927 29

1010 11

114 4

120 1

131 1

141 1

Over 14 0 0

Total2608 2608


Random events

Regular events

Clumped events

The Poisson Distribution0.1

00.20.40.60.8

1

0.5

00.20.40.60.8

1

1

00.20.40.60.8

1

2

00.20.40.60.8

1

6

00.20.40.60.8

1

Review of Discrete Probability Distributions

If X is a discrete random variable,

What does X ~ Bin(p, n) mean?

What does X ~ Poisson(λ) mean?

The Expected Value of a Discrete Random Variable

nn

n

iii papapapaXE

...)( 22111

The Variance of a Discrete Random Variable

22 )()( XEXEX

n

i

n

iiiii paap

1

2

1

Continuous Random Variables

If X is a continuous random variable, then X has an infinitely large sample space

Consequently, the probability of any particular outcome within a continuous sample space is 0

To calculate the probabilities associated with a continuous random variable, we focus on events that occur within particular subintervals of X, which we will denote as Δx

Continuous Random Variables

dxxxfXE

xxfxXE in

ii

)()(

)()(1

xxfxXP ii )()(

The probability density function (PDF):

To calculate E(X), we let Δx get infinitely small:

Uniform Random Variables

Defined for a closed interval (for example, [0,10], which contains all numbers between 0 and 10, including the two end points 0 and 10).

0

0.1

0.2

0 1 2 3 4 5 6 7 8 9 10X

P(X)

Subinterval [5,6]Subinterval [3,4]

otherwise

xxf,0

100,10/1)(

The probability density function (PDF)

Uniform Random Variables

2/)()( abXE

For a uniform random variable X, where f(x) is defined on the interval [a,b] and where a<b:

12)()(2

2 abX

otherwise

bxaabxf,0),/(1)(

The Normal DistributionOverview

Discovered in 1733 by de Moivre as an approximation to the binomial distribution when the number of trials is large

Derived in 1809 by Gauss

Importance lies in the Central Limit Theorem, which states that the sum of a large number of independent random variables (binomial, Poisson, etc.) will approximate a normal distribution

Example: Human height is determined by a large number of factors, both genetic and environmental, which are additive in their effects. Thus, it follows a normal distribution.

Karl F. Gauss (1777-1855)

Abraham de Moivre (1667-1754)


A continuous random variable is said to be normally distributed with mean and variance 2 if its probability density function is

f(x) is not the same as P(x) P(x) would be virtually 0 for every x because

the normal distribution is continuous However, P(x1 < X ≤ x2) = f(x)dx

f (x) =

12

(x )2/22

e

x1

x2


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

x

f(x)


Mean changes Variance changes

The Normal DistributionLength of Fish

A sample of rock cod in Monterey Bay suggests that the mean length of these fish is = 30 in. and 2 = 4 in.

Assume that the length of rock cod is a normal random variable X ~ N( = 30 , =2)

If we catch one of these fish in Monterey Bay, What is the probability that it will be at least

31 in. long? That it will be no more than 32 in. long? That its length will be between 26 and 29

inches?


What is the probability that it will be at least 31 in. long?

0.00

0.05

0.10

0.15

0.20

0.25

25 26 27 28 29 30 31 32 33 34 35

Fish length (in.)


That it will be no more than 32 in. long?

0.00

0.05

0.10

0.15

0.20

0.25

25 26 27 28 29 30 31 32 33 34 35

Fish length (in.)


That its length will be between 26 and 29 inches?

0.00

0.05

0.10

0.15

0.20

0.25

25 26 27 28 29 30 31 32 33 34 35

Fish length (in.)

-6 -4 -2 0 2 40

1000

2000

3000

4000

5000

Standard Normal Distribution

μ=0 and σ2=1

Useful properties of the normal distribution

The normal distribution has useful properties: Can be added: E(X+Y)= E(X)

+E(Y) and σ2(X+Y)= σ2(X)+ σ2(Y)

Can be transformed with shift and change of scale operations

Consider two random variables X and Y

Let X~N(μ,σ) and let Y=aX+b where a and b are constants

Change of scale is the operation of multiplying X by a constant a because one unit of X becomes “a” units of Y.

Shift is the operation of adding a constant b to X because we simply move our random variable X “b” units along the x-axis.

If X is a normal random variable, then the new random variable Y created by these operations on X is also a normal random variable .

For X~N(μ,σ) and Y=aX+b

E(Y) =aμ+b σ2(Y)=a2 σ2

A special case of a change of scale and shift operation in which a = 1/σ and b = -1(μ/σ): Y = (1/σ)X-(μ/σ) = (X-μ)/σ This gives E(Y)=0 and σ2(Y)=1

Thus, any normal random variable can be transformed to a standard normal random variable.

The Central Limit Theorem

Asserts that standardizing any random variable that itself is a sum or average of a set of independent random variables results in a new random variable that is “nearly the same as” a standard normal one.

So what? The C.L.T allows us to use statistical tools that require our sample observations to be drawn from normal distributions, even though the underlying data themselves may not be normally distributed!

The only caveats are that the sample size must be “large enough” and that the observations themselves must be independent and all drawn from a distribution with common expectation and variance.

Log-normal Distribution X is a log-normal

random variable if its natural logarithm, ln(X), is a normal random variable [NOTE: ln(X) is same as loge(X)]

Original values of X give a right-skewed distribution (A), but plotting on a logarithmic scale gives a normal distribution (B).

Many ecologically important variables are log-normally distributed.

rep 1994

1600.0

1500.0

1400.0

1300.0

1200.0

1100.0

1000.0

900.0800.0

700.0600.0

500.0400.0

300.0200.0

100.00.0

300

200

100

0

Std. Dev = 183.79

Mean = 127.5

N = 765.00

SOURCE: Quintana-Ascencio et al. 2006; Hypericum data from Archbold Biological Station

LOGREP94

7.256.75

6.255.75

5.254.75

4.253.75

3.252.75

2.251.75

1.25.75

70

60

50

40

30

20

10

0

Std. Dev = 1.44

Mean = 4.00

N = 765.00

A

Log-normal Distribution

2/2emean2

2

2

2

1

eevariance

Exercise During class we will perform an

exercise in R allowing you and us to work with some of these probability distributions!

random variables and probability distributions

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