Class Notes (1,100,000)

US (480,000)

UPenn (2,000)

STAT (70)

STAT 101 (30)

Richard Waterman (10)

Lecture 11

# STAT 101 Lecture Notes - Lecture 11: Independent And Identically Distributed Random Variables, Chocolate Chip Cookie, Bernoulli DistributionPremium

Department

StatisticsCourse Code

STAT 101Professor

Richard WatermanLecture

11This

**preview**shows page 1. to view the full**4 pages of the document.**Stat 101 - Introduction to Business Statistics - Lecture 11: Random Variables

The Bernoulli random variable

● The Bernoulli r.v. is the simplest type of random variable you a can get (apart from a

constant).

● It forms a building block for many other random variables.A coin ﬂip is an example of a

Bernoulli trial.A Bernoulli rv takes on one of two values. Either a 0 or a 1.Sometimes we

call the outcomes either a failure or a success.Any random variable with a dichotomous

outcome, can be thought of

● as a Bernoulli.

● Examples

○ 1 Live/Die.

○ 2 Buy, don’t buy.

○ 3 Market goes up/market goes down.

○ 4 Employee stays on the job/ employee quits.

● You have to equate one level of the outcome to the number 1, and the other to 0.

● Denote the random variable with the letter B and associate 1 with a success and 0 with a

failure. Denote P(B = 1) as p.

● E(B) = 0 × (1 − p) + 1 × p = p.

● Facts:

○ E(B) = p.

○ Var(B) = p(1 − p).

○ sd(B) = sqr root (p(1 − p))

Illegal downloading example

● Background: if an unauthorized work is downloaded, but the download originates outside

the US, it becomes a legal matter as to whether or not US law applies.

● A single download can be thought of as a Bernoulli trial.Equate 0 with the event that the

download happens inside the US.

● Equate 1 with the event that the download happens outside the US.

● The probability that the download happens outside the US is 0.1.

● We will now consider a sequence of downloads.

● Sequence of downloads:

○ The sum counts the total number of downloads from outside the US,because

each element in the sum is either a 1 or a 0.

○ Now assume that the sequence is iid, that is, independent and identically

distributed.

○ This means that the outcome of each event has no impact on any others, and

they are all Bernoulli trials with the same probability of“success”, in this case 0.1.

● The number of successes in n iid Bernoulli trials is called a Binomial random variable.

○ Y = B1 + B2 + · · · + B100.

● Using the formulas from slide 6 and the facts about the mean and variance of a Bernoulli

random variable it follows that

○ E(Y ) = np.

○ Var(Y ) = np(1 − p).

○ sd(Y ) = sqr root(np(1 − p))

###### You're Reading a Preview

Unlock to view full version

Subscribers Only