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Stats

random variable (RV)

is a function

i.e. number of coin tosses till H

w: H,TH,TTH, ...

x = 0,1,2...

X: w --> x: H --> 1, TH --> 2

Stats

probability mass function (PMF)

probability of having getting a heads on 1, 2, . . . tosses

(every discrete random variable X has it associated with it)

Stats

probability distribution functions (PDFs)

Continuous RVs has it

Stats

cumulative distribution function

cumulative probability of getting a heads in 1 or less tosses; 2 or less tosses

(discrete case)

Stats

binomial random variable

probability of x successes out of n is deﬁned by the pmf

*toss a coin n = 10 timessuccess and failure, each with probability θ and (1 − θ)*

Stats

What’s the probability of x or fewer successes, where x is some number between 0 and 10?

CDF function pbinom

Stats

What is the probability of getting exactly x successes?

answer from (a) the cdf, or (b) the pmf

Stats

define the discrete binomial random variable X

- function X : S → R, where S is the set of outcomes (i.e.,

outcomes are ω ∈ S) - X(ω) = x, and SX is the support of X (i.e., x ∈ SX)
- A PMF is deﬁned for X
- A CDF is deﬁned for X

Stats

associated R function

- cdf

- dnorm
- pnorm

Stats

The area under curve between +infty and -infty

pnorm(Inf)-pnorm(-Inf)

Stats

The area under curve between 2 and -2

pnorm(2)-pnorm(-2)

Stats

The area under curve between 1 and -1

pnorm(1)-pnorm(-1)

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