Probability distributions

Example

Variable G denotes the population in which a mouse belongs

G=1 : mouse belongs to population 1 G=2 : mouse belongs to population 1

Probabilities for the two alternatives define a probability distribution of G

P(G=1)=0.833 P(G=2)=0.167

…if the sum of the probabilities is equal to 1:

P(G=1)+P(G=2)=0.833+0.167=1

Probability distribution as a function

Probability distribution may be defined by a set of probabilities for the alternative values of a variable

Or by a function which assigns the probabilities to alternatives This is especially useful when there are

many alternatives The function usually has one or more

parameters, which control how the probability is distributed to different values

Example : Binomial distribution

x :Number of heads in 10 tosses of a coin Parameter N: number of tosses Parameter p: probability of heads in each trial

x | N,p ~ Bin(N,p)

P(x=k |N,p) ={ N!/(k!(N-k!)) } pk(1-p)N-k

Binomial distribution for the number of heads in 10 tosses of a fair coin

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

x

P(x

|N=

10

,p=

0.5

)

Continuous variables?

Infinite number of possible values between any two possible values.

->probability of any particular value = 0 There is probability density for each value: the

“height” of probability mass at that point There is probability between two points, found by

integration Practical calculations:

establish a dense grid of values at which to evaluate the probability density

Normalise the density by the sum of the grid: approximation of the amount of probability around each grid point

Example: Normal distribution

Possible values: all real numbers Parameter : Mean of the probability mass,

center of gravity Parameter 2 : variance of the probability

mass, controls the spread of the probability

Probability density of x

p(x=k| , 2 )=((22) -1/2 )exp{(k- )2 / 22 }

Normal distribution

0

0.005

0.01

0.015

0.02

0.025

10

0

12

0

14

0

16

0

18

0

20

0

22

0

24

0

26

0

28

0

30

0

32

0

34

0

x

p(x

|mu

=2

00

,sig

ma

=2

0)

Describing the probability distribution

Mean Variance Standard deviation Median and other percentiles Mode Coefficient of variation

Cumulative distribution

0

0.2

0.4

0.6

0.8

1

1.21

00

12

0

14

0

16

0

18

0

20

0

22

0

24

0

26

0

28

0

30

0

32

0

34

0

k

P(x

<k

|mu

=2

00

,sig

ma

=2

0)

Exercise 3

Make a graph showing the probability density of a Normal distribution with mean = 100 and standard deviation of 10. Evaluate the density at values 50,55,60,65,…,150

Using the grid approximation, calculate the following statistics of the distribution Mean Variance Standard deviation Coefficient of variation

Exercise 3 continues

By using the grid approximation, calculate the cumulative distribution of the previously defined normal distribution

Use the graph to determine the following statistics Median 5% percentile 95% precentile