3. Discrete RVs

date

Jun 21, 2023

slug

mat2017-3

author

status

Public

3. Discrete RVs

Probability mass function

Expectation

Bernoulli

indicator random variable of an event A

Parameter:

PMF:

Mean:

Variance:

Binomial

n independent Bernoulli trials, X be the ‘# of success’

Parameter:

PMF:

Mean:

Variance:

, , and X and Y are independent, then

Random walk

Y be the particle’s position after n steps

PMF:

Hypergeometric

# of success when sampling w/o replacement

Parameter:

PMF:

Mean:

Variance:

, then the conditional distribution of X is Hypergeometric

, , then

Mass Cash lottery

In each day that the Mass Cash lottery is run in Massachusetts, 5 of the integers from 1 to 35 are chosen (randomly and without replacement).

10 weeks of the lottery, every number picked at least once:

Indicator RV

for an event A is defined to be 1 if A occurs and 0 otherwise: Bernoulli

Card number matching problem

Haribo

: average # of bags that the fist students get in total

: average # of students who get at least one bag

Birthday matching problem

: expected # of birthday matches

: expected # of pairs of people with the same birthday

Geometric

# of failure before the first successful trial

Parameter:

PMF:

Mean:

Variance:

First Success

Mean:

Variance:

Coupon collector

Suppose there are n types of toys, which you are collecting one by one, with the goal of getting a complete set. When collecting toys, the toy types are random (as is sometimes the case, for example, with toys included in cereal boxes or included with kids’ meals from a fast food restaurant). Assume that each time you collect a toy, it is equally likely to be any of the n types. What is the expected number of toys needed until you have a complete set?

At least one boy and at least one girl

: add for first kids born

Negative Binomial

# of failures before the r-th success

Parameter:

PMF:

Mean:

Variance:

Poisson

Parameter:

PMF:

Mean:

Variance:

X is independent of Y, then conditional distribution of X given

, , and X and Y are independent, then

, , then

Website

The owner of a certain website is studying the distribution of the number of visitors to the site. Every day, a million people independently decide whether to visit the site, with probability p = 2 ⇥ 10−6 of visiting. Give a good approximation for the probability of getting at least three visitors on a particular day.

, then

Raindrop

Raindrops are falling at an average rate of 20 drops per square inch per minute. What would be a reasonable distribution to use for the number of raindrops hitting a particular region measuring 5 inches2 in t minutes? Why? Using your chosen distribution, compute the probability that the region has no rain drops in a given 3-second time interval.

20 per inch^2, 1 min → 5 inch^2, 1/20 min

Law school

There are n students at a certain school, of whom X ⇠ Bin(n, p) are Statistics majors. A simple random sample of size m is drawn (“simple random sample” means sampling without replacement, with all subsets of the given size equally likely).