Probability Distributions, Population and Samples

Question 1

Bernoulli distribution:
- what does it describes?
- what is the mean?
- what is the variance?

Answer 1

• the number of success in 1 single trial.
• mean p
• variance: pq

Question 2

Binomial distribution:
- what does it describes?
- what is the mean?
- what is the variance?

Answer 2

• the number of success in a fixed number of trials, each with a probability of success p.
• mean np
• variance: npq

Question 3

Normal Distribution:
- is defined by 2 parameters
- indicated with X~N(… )
- key property
- 1,2,3 dev std

Answer 3

it is defined by the mean and the variance: X ~N(mu, o^2).
- it is symmetric.
- 68% of data falls within 1 dev std of the mean; 95% within 2 dev std; 99.7% within 3 dev std.

Question 4

Standard Normal Distribution:
- it is a special case of … with mean … and variance …
- it is indicated with X~N( … )
- any normal distribution can be standardized, how? What do we obtain and what does it represents?
- formula for the standardization

Answer 4

• special case of normal distribution. X ~N(0,1).
• any normal distribution can be standardized: we take a value from the normal d., we standardize it and we get the z score which represents the number of dev std a value is from the mean.

z= x-mu / dev.std.

Question 5

Population and sample:
- what is the population and how its size is represented.
- what is the sample and how its size is represented.

Answer 5

• the entire group we want to study
• population size (N)
• a sample is a subset of the population
• sample size (n)

Question 6

When is a sample a good representation of the population?

3 main aspects of a good sample:
- random sampling.
- large sample size;
- independent, identically distributed samples.

Answer 6

When it accurately reflects the characteristics and variability of the population from which it is drawn.

• random sampling: each member of the population has an equal chance to be selected, otherwise the sample would not reflect the variability of the population.
• large sample size: the bigger the sample size and the smaller the variability of the sample meaning that tha sample statistics will be closer to the population statistics.
• independent, identically distributed samples: each observation is independent of the others and has the same probability distribution as the population.

Question 7

Point estimation:
- what process is it?
- what is the result?

3 characteristic of a good estimator:
- unbiased;
- consistency;
-efficiency.

Answer 7

• is the process of estimating an unknown parameter of the population using sample data.
• it gives a single point as an estimate of the unknown parameter of the pop.
• unbiased: the expected value of the estimator should be equal to the true parameter.
• consistency: as the sample size increases, the estimator should converge to the true parameter.
• efficiency: among the unbiased estimators, the one with the smallest variance is preferred.

Question 8

What is the maximum likelihood estimation (MLE)? How does it work?

Answer 8

Is the method for estimating parameters of a statistical model.

It works by finding the parameter values that maximize how likely it is to observe the given sample data given certain parameter values (likelihood function).

Question 9

What is the mean?
- how is it calculated in the population?
- how is it calculated in a sample?

What is the proportion?
- how is it calculated in the population?
- how is it calculated in a sample?

What is the variance?
- how is it calculated in the population?
- how is it calculated in a sample?

Answer 9

• measure of central tendency that represents the center of a set of values.
• mu = summation(Xi) / N
• x trattino = summation(xi) / n
• the relative amount of a sub-group compared to the entire population.
• p = X / N
• p cappello = x / n
• measure of how spread out the data is, how far points are from the mean.
• o^2 = summation((Xi-mu)^2)/N;
• s^2 = summation((xi-xtrattino)^2)/n.

Question 10

Central Limit Theorem:
- pre-requisite (and rule of thumb)
- what does it say?
- in particular….

Answer 10

• n is large an enough (n>= 30)
• regardless of the population’s original distribution, the distribution of the sample means will tend to become approximately normal.
• if you repeatedly take samples from a population and calculate the mean of each sample, the distribution of those means will resemble a normal distribution as the sample size increases.

Question 11

Distribution of the sample mean:
- central limit theorem
- what is the mean of the sample means? E[xtrattino]?
- what is the standard error of the sample means? 2 possibilities
- as the sample size increases what happen to devstd?

Answer 11

• central limit theorem: the distribution of the sample means is approximately normal.
• E[xtrattino] = mu: the expected value of the sample mean is the population mean.
• 1. if we know the variance of the population, standard error = devstd o/sqrt(n).
     
     2. if we do not know the variance of the population, standard error = devstd s/sqrt(n).
• As the sample size n increases, the standard error decreases.

Question 12

Distribution of the sample proportion:
- what is the mean of the sample proportion? E[pcappello]?
- what is the standard error of the sample proportion?
- as the sample size increases what happen to devstd?

Answer 12

• E[pcappello] = p: the expected value of the sample proportion is the population proportion.
• standard error = sqrt(pq/n).
• As the sample size n increases, the standard error decreases.