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Lecture 11

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


Department
Statistics
Course Code
STAT 101
Professor
Richard Waterman
Lecture
11

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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))
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