3.1 Statistical Distributions

Random Variable

• Observation = Variable Outcome = Random Variable

• Examples: – Weight/Size of animals

– Animal surveys: detection outcomeseen/not seen?

Probability Distributions• Used to describe patterns of variability in outcomes

(e.g., # of times detected, weight)

• Depends on the attributes of a random variables (RV)Ex: Weight:

– Continuous– Necessarily positive– Most in narrow range with some large and small values

• Use mathematical function to describe variation in a set of values

Attributes of R.V. => Type of Distribution

• Discrete or continuous– Discrete: survival, reproductive status, litter size,

population size, sex, observed/not observed etc.– Continuous: weight, length, rainfall, temperature,

etc.• Range of possible values? (limited range?)• Mean – average value• Variance – spread in values• Skewness, kurtosis – shape of the variability

(skewed, Peak vs. Tails)

probability mass (density) function

• Discrete – the probability of each discrete value occurring (pmf)

• Continuous – probability of the value occurring within an interval (pdf)

• Must sum to one

Specification of a distribution

• Parameters: fixed quantities needed to describe the distribution

• Expected value (mean): E(X) – the weighted (Pr.) average over all possible values

• Variance: V(X) – averaged squared difference between expected value and each of the possible values

Standard Deviation =

Statistical distributions most

commonly used in our field

Bernoulli Distribution

• “Coin flip” - Possible values are or • Single parameter p – probability of getting a ‘success’

()

• E(X)= (1-p)*0 + p*1 = p

• Var(X) = p(1-p)

)p - (1 p = p) | f(x x - x 1

• let p = 0.8

• f(x=0|p=0.8) = 1-p = 0.2 and f(x=1|p=0.8) = p = 0.8

• E(x) = 0.2*0 + 0.8*1 = 0.8

• Variance = 0.2*(0.8^2) + 0.8*(0.2^2) = 0.16

)p - (1 p = p) | f(x x - x 1

Example

Binomial (Multiple Bernoulli trials)• Consider groups of Bernoulli r.v.’s (N = 3)• Possible outcomes 000,100,010,001,011,101,110,111

• 0/3 : (1-p)3 000 (1 way)• 1/3 : p(1-p)2 100, 010, 001 (3 ways)• 2/3 : p2 (1-p) 011, 101, 110 (3 ways)• 3/3 : p3 111 (1 way)

• Combinations: number of ways of “choosing” the outcomes

• Total # of successes is a binomial random variable

Binomial Distribution– Parameters: bin(n, p)

• n independent trials• p = probability of success

at each trial

– Possible outcomes 0,1,2,….n• X= number of successes

– The same as n independent Bernoulli(p)– Combination term occurs in the function (X are 1, n-X are 0)

– E(X) = np– Var(X)= p(1 – p)n

, x n - xnf(x | p n) = (1 - pp )

x

)!(!!

xnxn

x

n

Example• n = 10 nesting sea turtles captured in year 1

• X = number that are captured the following year

• S = probability caught in the next year (survival?)

1010Pr( | , 10) - xxX x S n = (1 - S )Sx

0 1 2 3 4 5 6 7 8 9 100.0 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0000.1 0.349 0.387 0.194 0.057 0.011 0.002 0.000 0.000 0.000 0.000 0.0000.2 0.107 0.268 0.302 0.201 0.088 0.026 0.006 0.001 0.000 0.000 0.0000.3 0.028 0.121 0.234 0.267 0.200 0.103 0.037 0.009 0.001 0.000 0.0000.4 0.006 0.040 0.121 0.215 0.251 0.201 0.112 0.043 0.011 0.002 0.0000.5 0.001 0.010 0.044 0.117 0.205 0.246 0.205 0.117 0.044 0.010 0.0010.6 0.000 0.002 0.011 0.043 0.112 0.201 0.251 0.215 0.121 0.040 0.0060.7 0.000 0.000 0.001 0.009 0.037 0.103 0.200 0.267 0.234 0.121 0.0280.8 0.000 0.000 0.000 0.001 0.006 0.026 0.088 0.201 0.302 0.268 0.1070.9 0.000 0.000 0.000 0.000 0.000 0.002 0.011 0.057 0.194 0.387 0.3491.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000

x

S

Binomial distribution

n!

x! (n – x)!Sx (1–S)(n–x)Pr(X = x) =

Example

•A better way to think about the distribution of the number of turtles recaptured (x) will be

•where– S = survival – F = fidelity to population, given survival– g breeding probability, given faithful– p = probability captured, given breeder.

1010Pr( ) ( )x - xX x = S (1 - SF pF p )

xgg

Multinomial Distribution• Number of times each of > 2 discrete possible

outcomes occur in N trials• Example: rolling a die

6 possible outcomes ; πi = 1/6

• Parameters: N and πi’s• Function looks similar to the binomial –

combinatorial term and probability ) n , n , n , n , n , Pr(n 654321

N654321 n

6n5

n4

n3

n2

n1

654321

!n ! n ! n ! n ! n ! n

!

Examples, with N = 3 trials:

• Probability of getting three 1’s –– 3!/3!0!0!0!0!0! = 1

Result: P(3,0,0,0,0,0) = 1 x (1/216) = 1/216

• Probability of one 1, one 4,and one 6– 3!/1!0!0!1!0!1! = 6

Result: P(1,0,0,1,0,1) = 6 x (1/216) = 6/216

216/1)6/1( 331

216/1)6/1( 316

14

11

N654321 n

6n5

n4

n3

n2

n1

654321

!n ! n ! n ! n ! n ! n

!

Poisson Distribution

• Often used for counts (e.g., number of eggs in a nest, # of animals in a plot)

• Possible values are all non-negative integers• λ is the only parameter• E(x) = λ and V(x) = λ ( | ) / !xf x e x

Normal Distribution• Extremely common and useful distribution, with

location (μ) and shape (σ²) parameters

• Why is this useful?– Many variables are normally distributed– Central limit theorem: For large n’s, means tend to be

normally distributed, even if not originally normal– Estimates of parameters intuitive

• E(x) = μ• Var(x) = σ2

. - x21-

21=) , | f(x

2

exp ) ,N( 2

• Standard normal

• pdf:

Any normal can be converted to a standard normal= Standardization of R.V.: (X – μ)/σ

.Z21-

21=) , | f(Z 2

exp10

) ,N( 10

Normal Distribution

Other useful distributions

• Discrete:– Negative binomial (less restrictive than Poisson)

•Continuous– Χ2 and F distributions – used for test statistics– Uniform – all values in an interval have same

probability– Beta – flexible distribution for values between 0 and 1– Gamma – flexible distribution for values ≥ 0

To remember

• Observations = Random Variables

• Distribution => Describes Variability

• Parameters describes distribution

Uses of Probability Distributions• Frequency of possible outcomes: Frequency of values (observed)

for samples from a population (e.g., weights, # survived and dead)

• Probability of specific outcomes: e.g., Pr. of observing a marked animal during 3 consecutive samples

• Uncertainty in prediction: e.g., the expected number of animals that will survive over the next winter is between 70% and 80%

• Uncertainty in knowledge: Description of our “belief” about the value of a parameter (e.g., the expected proportion of males at birth is almost certainly between 0.45 and 0.55)