STAT 101 Lecture Notes - Lecture 11: Independent And Identically Distributed Random Variables, Chocolate Chip Cookie, Bernoulli DistributionPremium
Course CodeSTAT 101
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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
● 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.
○ 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.
○ 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
○ 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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