Download Lecture 4: Random Variables and Distributions and more Lecture notes Probability and Statistics in PDF only on Docsity!

Lecture 4: Random

Variables and Distributions

Goals

• Working with distributions in R
  • Overview of discrete and continuous

distributions important in genetics/genomics

• Random Variables

Two Types of Random Variables

• A discrete random variable has a

countable number of possible values

• A continuous random variable takes all

values in an interval of numbers

Probability Distributions of RVs

Discrete

Let X be a discrete rv. Then the probability mass function (pmf), f(x), of X is: f ( x ) = P(X = x), x ∈ Ω 0, x^ ∉^ Ω

Continuous

P ( a " X " b ) = f ( x ) dx a b

Let X be a continuous rv. Then the probability density function (pdf) of X is a function f(x) such that for any two numbers a and b with a ≤ b: a b A a

Using CDFs to Compute Probabilities

Continuous rv: ! F ( x ) = P ( X " x ) = f ( y ) dy #$ x % pdf (^) cdf P ( a " X " b ) = F ( b ) # F ( a )

Expectation of Random Variables

Continuous

X = E [ X ] = x " f ( x ) dx #$ $

The expected or mean value of a continuous rv X with pdf f(x) is:

Discrete

Let X be a discrete rv that takes on values in the set D and has a pmf f(x). Then the expected or mean value of X is:

X

= E [ X ] = x " f ( x )

x # D

Example of Expectation and Variance

• Let L 1 , L 2 , …, L n be a sequence of n nucleotides and define the rv X i : 1, if L i = A 0, otherwise X i
• pmf is then: P(X i = 1 ) = P(L i = A) = p A P(X i = 0 ) = P(L i = C or G or T) = 1 - p A
• E[X] = 1 x p A

• 0 x (1 - p A ) = p A

• Var[X] = E[X - μ] 2 = E[X 2 ] - μ 2 = [ 1 2 x p A

• 0 2 x (1 - p A )] - p A 2 = p A (1 - p A )

The Distributions We’ll Study

1. Binomial Distribution
2. Hypergeometric Distribution
   3. Poisson Distribution
      4. Normal Distribution

Binomial Distribution

! P { X = x } = (^) ( ) p x ( 1 " p ) n n " x x

pmf:

E(x) = np

cdf:

P { X " x } = ( ) p

y

( 1 # p )

n # y y = 0 x $ n y

Var(x) = np( 1 -p)

Binomial Distribution: Example 1

• A couple, who are both carriers for a recessive

disease, wish to have 5 children. They want to know

the probability that they will have four healthy kids

! P { X = 4 } = (^) ( )0. 4 " 0. (^5) 1 4

0 1 2 3 4 5 p(x)

Hypergeometric Distribution

• Population to be sampled consists of N

finite individuals, objects, or elements

• Each individual can be characterized as a

success or failure, m successes in the

population

• A sample of size k is drawn and the rv of

interest is X = number of successes

Hypergeometric Distribution

• Similar in spirit to Binomial distribution, but from a finite population without replacement 20 white balls out of 100 balls If we randomly sample 10 balls, what is the probability that 7 or more are white?

Hypergeometric Distribution

• Extensively used in genomics to test for “enrichment”: " = Number of annotated genes Number of genes of interest Number of genes with annotation Number of genes of interest with annotation

Poisson Distribution

• Useful in studying rare events
• Poisson distribution also used in situations

where “events” happen at certain points

in time

• Poisson distribution approximates the

binomial distribution when n is large and p

is small