Module 5: Special Distributions, the Sample Mean, and the Central Limit Theorem

For which value(s) 𝑝∈[0,1] does Bernoulli variable with probability of success 𝑝 have minimum variance?

0, 1

For which value(s) 𝑝∈[0,1] does Bernoulli variable with probability of success 𝑝 have maximum variance?

0.5

What are reasonable ways to estimate 𝜃?

Compute the maximum (𝑛𝑡ℎ order statistic) of the sample or Compute the sample mean and multiply by 2

The Central Limit Theorem (CLT) implies that were one to draw 𝑛 samples 𝑋1,...𝑋𝑛 independently and identically, then for reasonably large 𝑛...

Each 𝑋𝑖 need not be approximately normally distributed, but the sample mean 𝑋¯=∑𝑖𝑋𝑖/𝑛 will be approximately normally distributed.

In the context of the exponential distribution, what is meant by memorylessness?

If x describes the waiting time for some event, then the probability distribution of 𝑥 at 𝑡=0 is the same as the probability distribution of 𝑥 at time 𝑡=1 or 𝑡=100 when the event has not occurred, for example.

What is true about a sample mean?

It can be described as the arithmetic average of 𝑛 random variables from a random sample of size 𝑛. It can be described as the arithmetic average of the realizations of 𝑛 random variables.

For an i.i.d. distribution, how does the expectation of sample mean of 𝑛 random variables drawn from this distribution vary with 𝑛?

No effect on the expectation of the sample mean

What are the requirements for a series of events to be effectively modeled according to the Poisson distribution?

Occurrences of the event must be countable and measurable Each of the events are independent The average frequency of occurrences is known for a certain time period

What is the appropriate interpretation of the Binomial distribution?

The number of successes in a sequence of n success/failure trials, each of which has the same probability of success, 𝑝

Suppose a soccer team's goal-scoring 𝑋 in each of their games follows a (fixed) Poisson distribution. What does the 𝑃(𝑋=2𝜆) capture?

The probability that in a game, the soccer team scores twice their expected number of goals

Bernoulli Distribution

a discrete data distribution used to describe a population of binary variable values

True or False: Estimators are the realizations of applying estimates to random samples.

false

True or False: To prove an estimator is unbiased, you need to know the value of the parameter it is trying to estimate.

false

True or False: If you have a set of i.i.d normal random variables, then any linear combination of these variables will follow a uniform distribution.

false - any linear combination will also follow a normal distribution

True or False: If 𝑋1,...,𝑋𝑛 are i.i.d. and 𝑋𝑖∼𝑁(𝜇𝑖,𝜎2𝑖) then 𝑌=∑𝑖𝑋2𝑖∼𝑁(∑𝑖𝜇2𝑖,∑𝑖𝜎2𝑖)

false - only true for linear combinations

An estimator is unbiased if

its expected value is equal to the true population parameter it is supposed to be estimating 𝐸(𝜃̂ )=𝜃 for all 𝜃

when you increase sample size, the histogram...

narrows - more values of the mean

Suppose that you have a Bernouilli variable X with some probability of success given by 𝑝 and some probability of failure given by 𝑞. The mean of 𝑋 is given by:

p

For all families of distributions, estimation is trying to determine the specific ______ of a distribution.

parameter

________ is the process of subtracting the mean of a distribution and dividing by the square root of its variance.

standardization

Fill in the blanks with the correct interpretation: In the period of length _______, we can expect there to be ________ arrivals, or occurrences of the event.

t ; 𝜆 t ; 𝛾*t

True or False: Suppose you are able to generate random numbers from a normal distribution (𝜇,𝜎2) of unknown mean 𝜇. If 𝜇̂ is the random variable whose realizations are the individual numbers generated, 𝜇̂ is an unbiased estimator for 𝜇.

true

True or False: Taking a linear transformation of a normally-distributed random variable generates a normally-distributed random variable. In other words, if 𝑋1 is normally-distributed and 𝑋2=𝑎+𝑏∗𝑋1 for 𝑏≠0, then 𝑋2 is also normally-distributed.

true

True or False: When the sample mean is defined as the arithmetic average of 𝑛 random variables from random sample of size 𝑛, the sample mean will also be a random variable.

true

True or False: You can uniquely identify a given distribution if you know the family of distributions it is from (ex. Normal, uniform etc.) and the value of the relevant parameters for that family.

true

Per the notational convention, what is the usual relationship between 𝜃 and 𝜃̂ ?

𝜃 usually refers to a parameter relevant to the underlying distribution, while 𝜃̂ refers to its estimation in a finite sample

Which of the following are the typical notations used for parameters of the normal distribution?

𝜇 and 𝜎2

For a sample of size 𝑛 from an i.i.d distribution with variance 𝜎2, which of the following expressions is the variance of the sample mean?

𝜎^2/𝑛