Lecture 2 Probability DistributionsTheophanis Tsandilas

Comment on measures of dispersion

Why do common measures of dispersion (variance and standard deviation) use sums of squares:

instead of sums of absolute residuals:nX

i=1

|xi � µ̂|

nX

i=1

(xi � µ̂)2

Comment on measures of dispersion

Working with absolute values can be difficult, but this is not the main reason.

The measure of central tendency that minimizes the sums of absolute differences is the median, not the mean.

And since the mean is the prevalent measure of central tendency, we commonly use sums of squares.

However, for statistical methods that rely on medians, sums of absolute differences can be more appropriate.

R code

Consider the population of all possible outcomes when throwing two dice. How probable is each sum S of counts from the two dice?

What is a probability distribution?

The probability distribution provides the probability of occurrence of all possible outcomes in an experiment

Probability distribution of a population

It is generally not known. However, we may have sufficient information about this distribution to meet the goals of our research.

Statistical modeling: It is possible to rely on a small set of probability distributions that capture the key characteristics of a population.

We can then infer the parameters of these model probability distributions from our sample.

Why? We expect that the sample contains some information about the population from which it was sampled.

Example distributions

Height: Women vs. Men

Example distributions

US Income distribution (older than 2013)

Example distributions

Source: https://medium.com/runkeeper-everyone-every-run/how-long-till-the-finish-line-494361cc901b

Half Marathon and Marathon race finish time

Example distributionsDistribution of most frequent English words

http://norvig.com/mayzner.html

The population (and hence its samples) contain discrete values, either finite or infinite in number

e.g., {-3, 1, 0, 1, 2}, {‘blue’, ‘brown’, ‘green’}, or {1, 2, 3, ...}

The probability of a value x in a population can be expressed as a function: f(x) = p(X = x) (the probability of random variable X taking value x)

The function f(x) is called a probability mass function (pmf)

Discrete probability distributions

Discrete probability distributions

This is a discrete probability distribution

Discrete probability distributions

Probabilities should sum to 1:

The expected value (or mean):

The mode is the most frequent value: which one?

The median is the middle value: which one?

X

x2S

f(x) = 1

E(x) =X

x2S

f(x)x

= 21

36+ 3

1

18+ 4

1

12+ 5

1

9+ 6

5

36+ 7

1

6+ 8

5

36+ 9 ⇤ 1

9+ 10

1

12+ 11

1

18+ 12

1

36= 7

Symmetrical probability distributions

When the mean coincides with the median

The above is a symmetrical, unimode distribution

The binomial distribution

Consider a population containing two values: 1 and 0 Let’s set the probability of 1 to Pr(1) = .75 and the probability of 0 to Pr(0) = .25.

A single sample (n = 1) from such a population is known as a Bernoulli trial.

a coin flipping trial with a fair coin is a Bernoulli trial with Pr(Head) = .5 and Pr(Tail) = .5

If we perform n independent Bernoulli trials, their outcomes will follow a binomial distribution.

The binomial distribution

If a random variable X has a binomial distribution, we write:

where n and P are the parameters of the distribution: • n is the number of Bernoulli trials • P is the chance (probability) of success

If we know n and P, we can fully describe the distribution

X ⇠ B(n, P )

The binomial distribution

Probability Mass Function (pmf)

f(x;n, P ) =

✓n

x

◆P

x(1� P )n�x

The number of possible ways that n Bernoulli trails lead to x successes

The probability of exactly x successes

The probability of exactly n - x failures

The binomial distributionConsider the distribution of errors for 10 trials, when for each trial, errors occur with a probability of 40%

R code

The binomial distribution

Cumulative Distribution Function (cdf)

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Continuous distributions

Not restricted to specific values. They can take any value between the lower and upper bound of a population (of course, populations can be unbounded).

The probability of any particular value is zero. Probabilities can only be obtained for intervals (i.e., a range of values):

where f(x) is the probability density function (pdf). It provides the relative (rather than absolute) likelihood that a random variable X takes the value x.

Continuous distributions

Pr(a X b) =

Z b

af(x)dx

Pr(�1 X 1) =

Z 1

�1f(x)dx = 1

Continuous distributions

Mode: value of highest peak

Median: value that divides the area exactly in half

Mean: µ =

Z 1

�1xf(x)dx

The normal distribution

Also known as the Gaussian distribution

The normal distribution

Symmetrical, unimodal and continuous

Can be derived as a sum of an infinite number of independent random variables

Thus, it is appropriate when data arise from a process that involves adding together the contributions from a large number of independent, random events.

Example

The human height can be considered as the outcome of many independent random genetic and environmental influences

Normal distribution parameters

A normal distribution can be fully described by two only parameters: its mean μ and its variance σ2

A normally distributed variable X can be written as

Its probability density function (pdf) is as follows:

X ⇠ N(µ,�2)

f(x;µ,�2) =1p2⇡�2

e

� (x�µ

2)

2�2

Example

The following normal distribution has mean μ = 100 and a standard deviation σ = 15

Standard normal distribution

It is the normal distribution with a mean equal to 0 and a standard deviation (also variance) equal to 1:

The standard normal distribution is often abbreviated to z. It is frequently used to simplify working with normal distributions.

z ⇠ N(0, 1)

Standard normal distribution

Reading a normal distribution

Sampling distribution of a statistic

It is the distribution obtained by calculating the statistic (e.g. the mean) from an infinite number of independent samples of size n

Example

An experiment measures the time it takes n = 10 people to visually locate a target on a computer screen.

The same experiment is repeated a large (or infinite) number of times, where each time, we draw a new sample of size n. For each experiment, we compute the mean time:

Experiment 1: M = 11.4 sExperiment 2: M = 12.6 sExperiment 3: M = 10.2 s...

What’s the distribution of these mean values?

Sampling distribution of a statistic

Such distributions are interesting as they determine the probability of observing a particular value of the statistic, e.g., the mean.

It is often very different than the distribution of the data used to calculate the statistic.

distribution of the data ≠ sampling distribution of their means

Sampling distribution of the mean

Its mean value is also the mean (expected value) of the original population the samples were drawn from

Its standard deviation (SD) is known as the standard error of the mean (SEM)

The central limit theorem (CLT)

States that the sampling distribution of a statistic approaches the normal distribution as n approaches infinity.

It applies to statistics computed by summing or averaging quantities (means, variances) but not to standard deviations (squared root of an average)

The central limit theorem (CLT)

States that the sampling distribution of a statistic approaches the normal distribution as n approaches infinity.

It applies to statistics computed by summing or averaging quantities (means, variances) but not to standard deviations (squared root of an average)

central = fundamental to probabilities and statistics limit = refers to a limit condition n ! 1

Practical importance of the CLT

If the size of the sample is sufficiently large, then the sampling distribution of the statistic will be approximately normal

(no matter what the distribution of the original population was)

But which sample size is sufficiently large?

Sampling from normal distributions

If the original population is normal, then the CLT will always hold, even if the sample size is as low as n = 1

The further the original population moves away from a normal distribution, the larger the sample size n should be

Sampling from binomial distributions

n = 10, P = .15

Frequency

0 2 4 6 8

010000

25000

n = 30, P = .15

Frequency

0 2 4 6 8 12

05000

15000

n = 100, P = .15

Frequency

5 10 20 30

05000

15000

n = 10, P = .35

Frequency

0 2 4 6 8 10

010000

20000

n = 30, P = .35

Frequency

5 10 15 20

05000

10000

n = 100, P = .35

Frequency

20 30 40 50

05000

15000

Statistic of interest: Count of successes from n Bernoulli trials

R code

Sampling from exponential distributions

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

x

dexp(x)

n=10

Frequency

0.0 0.5 1.0 1.5 2.0 2.5

05000

10000150002000025000

n=30

Frequency

0.5 1.0 1.5 2.0

05000

10000

15000

20000

n=100

Frequency

0.6 0.8 1.0 1.2 1.4

05000

10000

15000

20000

Distribution of source population

Sampling distributions of the mean

Statistic of interest: Mean of a sample of n drawn from an exponential distribution

R code

What n is sufficiently large?

Several textbooks claim that n = 30 is enough to assume that a sampling distribution is normal, irrespective of the shape of the source distribution.

« This is untrue » [Baguley] There is no magic number to guarantee that.

Log-normal distribution

A random variable X is log-normally distributed if the logarithm of X is normally distributed:

X ⇠ LogN(µ,�2) () ln(X) ⇠ N(µ,�2)

Simple math with logarithms

logb(x) = a () b

a = x

logb(1) = 0 () b

0 = 1

logb(b) = 1 () b

1 = b

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If the base of the logarithm is equal to the Euler number e = 2.7182, we write:

ln(x) = loge(x)

Which we base to use is not important, but it is common to use e as a base.

Log-normal distribution

A common choice for real-world data bounded by zero e.g., response time or task-completion time

« The reasons governing frequency distributions in nature usually favor the log-normal, whereas people are in favor of the normal »

« For small coefficients of variation, normal and log-normal distribution both fit well. »

[ Limpert et al. 2001 ] https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

Sampling from lognormal distributions

0 1 2 3 4 5

0.0

0.2

0.4

0.6

x

dlno

rm(x

, mea

nlog

= 0

, sdl

og =

1)

n=10

Frequency

0 2 4 6 8 10 12 14

010000

30000

50000

n=30

Frequency

2 4 6 8

010000

20000

30000

40000

n=100

Frequency

1.0 1.5 2.0 2.5 3.0 3.5

05000

15000

25000

Distribution of source population

Sampling distributions of the mean

µ = 0,� = 1

Sampling from lognormal distributions

n=10

Frequency

0 2 4 6 8 10 12 14

010000

30000

50000

n=30

Frequency

2 4 6 8

010000

20000

30000

40000

n=100

Frequency

1.0 1.5 2.0 2.5 3.0 3.5

05000

15000

25000

n=10

Frequency

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

05000

10000

15000

20000

n=30

Frequency

-0.5 0.0 0.5

05000

10000

15000

20000

n=100

Frequency

-0.4 -0.2 0.0 0.2 0.4

05000

10000

15000

Before...

...and after applying a log transformation on the data

Sampling distributions of the mean

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Skewed distributions

Asymmetrical distributions are said to be skewed

The chi-square (χ2) distribution Consider a squared observation z2 drawn at random

from the standard normal (z) distribution

The distribution of z2 will follow a χ2 distribution with 1 degree of freedom (df)

The chi-square (χ2) distribution Consider a squared observation z2 drawn at random

from the standard normal (z) distribution

The distribution of z2 will follow a χ2 distribution with 1 degree of freedom (df)

-3 -2 -1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

x

Pro

babi

lity

dens

ity

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

1.2

x

Pro

babi

lity

dens

ity

Z Z2

The chi-square (χ2) distribution

A χ2 distribution with k degrees of freedom is the distribution of a sum of squares of k independent variables that follow a standard normal distribution

Q =kX

i=1

Z2i =) Q ⇠ �2(k)

The chi-square (χ2) distribution

Given the link between variances and sums of squares, the chi-square distribution is useful for modeling variances of samples from normal (or approximately normal) distributions.

The t distribution (Student’s t)

The sampling distribution of means for a normally distributed population

Useful when: the sample size is small, and

the population standard deviation is unknown

published by William Gosset (1908) under the pseudonym « Student »

The t distribution (Student’s t)

When the population standard deviation σ is unknown and is estimated from the unbiased variance estimate:

then, the resulting standardized sample mean has a t distribution with ν = n - 1 degrees of freedom.

�̂

2 =

nPi=1

(xi � µ̂)

n� 1

Note: We’ll further explain this later.

The t distribution (Student’s t)

A random variable X following a t distribution is denoted as:

X ⇠ t(⌫)

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4

x

Pro

babi

lity

dens

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tz

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4

x

Pro

babi

lity

dens

ity

tz

t(1) t(29)

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R distribution functions

dbinom(x, n, P) Provides the probability mass function for the binomial distribution B(n,P)

Examples:dbinom(4, 20, .2): It will return the probability of x = 4 successes for n = 20 Bernoulli trials with a P=.2 probability of success.

dbinom(c(1,2,3,4), 10, .2): It will return a vector with the probabilities of x = {1, 2, 3, 4} successes for n =10 Bernoulli trials with a P =.2 probability of success.

Binomial distribution

R distribution functions

pbinom(x, n, P) Provides the cumulative probability mass function for the binomial distribution B(n,P) Example: pbinom(4, 20, .2): It will return the cumulative probability up to x = 4 successes for n = 20 Bernoulli trials with a P = .2 probability of success.

Binomial distribution

R distribution functions

rbinom(size, n, P) It will generate a random sample of size size from the binomial distribution B(n,P) Example: rbinom(10, 20, .2): It will return a random sample of size = 10 from the binomial distribution B(n = 20, P = .2)

Binomial distribution

R distribution functions

dnorm(x, mean, sd) Provides the probability density function for the normal distribution with a mean value equal to mean and a standard deviation equal to sd.

Examples:dnorm(.2, 0, 1): It will return the relative likelihood of the value x = .2, for the standard normal distribution.

curve(dnorm(x, 100, 15), xlim = c(60, 140)): It will plot the probability density function from x = 60 to x = 140 for the normal distribution with mean = 100 and sd = 15.

Normal distribution

R distribution functions

pnorm(x, mean, sd) Provides the cumulative probability density function for the normal distribution with a mean value equal to mean and a standard deviation equal to sd.Example: pnorm(100, 100, 15): It will return the cumulative probability up to x = 100 for the normal distribution with mean = 100 and sd = 15. (What do you expect it to be?)

Normal distribution

R distribution functions

rnorm(size, mean, sd) It will generate a random sample of size size from the normal distribution with a mean value equal to mean and a standard deviation equal to sd.Example: rnorm(10, 0, 1): It will return a random sample of size = 10 from the standard normal distribution.

Normal distribution

R distribution functions

Distribution function (pmf or cdf)

Cumulative distr. function

Random sampling

Binomialdbinom(x, n, P) pbinom(x, n, P) rbinom(size, n, P)

Normaldnorm(x, mean, sd) pnorm(x, mean, sd) rnorm(size, mean, sd)

Log-normaldlnorm(x, mean, sd) plnorm(x, mean,

sd)rlnorm(size, mean, sd)

chi-squared dchisq(x, k) pchisq(x, k) rchisq(size, k)

Studentdt(x, ν) pt(x, ν) rt(size, ν)

Intro to Confidence Intervals

Statistical inference

The process of deducing the parameters of an underlying probability distribution from a sample

Four broad types: point estimation

interval estimation

hypothesis testing

prediction

Point estimates

How much informative is the following graph?

T1 T2 T3

Techniques

Mea

n Ti

me

(s)

01

23

4

Point estimates

A point estimate can be thought of as a « best guess » of the true population parameter

Descriptive statistics such a the sample mean or the median are examples of point estimates

Question: What are the point estimates of a population’s variance and standard deviation?

Point estimates

How much informative is the following graph?

A point estimate communicates no information about the uncertainty or quality of the estimate it provides

T1 T2 T3

Techniques

Mea

n Ti

me

(s)

01

23

4

Interval estimate

An interval estimate does not provide an exact value, but rather a range of values that the parameter might plausibly take.

Most common method: constructing a confidence interval (CI)

Confidence interval (CI)It specify a range of values that is expected to contain the true parameter value (but it may not)

It is associated with a confidence level, usually expressed as a percentage

e.g., 95% CI or 99% CI

true value

Formal interpretation of CIsClassical frequentists statistics view a probability as a statement about the frequency with which events occur in the long run.

Of the many 95% CIs that might be constructed, 95% are expected to contain the true population parameter. The other 5% may completely fail!

Informal interpretation of CIs

Formally speaking, a CI does not specify a probability range! A 95% CI does not necessarily contain the true population parameter with a 95% probability (or 95% confidence).

However, it is often reasonable to treat a CI as an expression of confidence or belief that it does contain the true value. See [Baguley] and [Cumming and Finch, 2005]

Attention: This view is not shared by everyone! It has received a lot of criticism by Bayesian statisticians.

Confidence level

A 100% CI will include the whole range of possible values

A 0% CI reduces to a point estimate

A 95% CI is the most common choice (by tradition)

alpha

If C is the confidence level of a CI, then:

C = 100 (1 - α)

where α (or alpha) represents the number of times that a C% CI is expected to fail:

If C = 95, the α = .05

Structure of a confidence interval

It is defined by two points that form its limits, i.e., its lower and upper bounds

It can be symmetrical, where the point estimate lies in the center of the CI

...or asymmetrical, where the point estimate is not at the center of the CI

Symmetrical CIs

The intervals can be described by the point estimate plus or minus half of the interval, e.g., 165 ± 6 cm

This half width of the interval is known as the margin of error (MOE)

Width of a CI

Depends on the confidence level e.g., 99% CIs are wider than 95% CIs

Also depends on the sampling distribution of the statistic

The smaller the sample size, the wider the sampling distribution

Small samples produce wide CIs

Example

Consider the sampling distribution of the mean for a normally distributed population (M = 100, SD = 10)

The sampling distribution becomes narrower as more samples are added.

n=10

Frequency

85 90 95 100 105 110 115 120

05000

10000

15000

20000

n=30

Frequency

85 90 95 100 105 110 115 120

05000

10000

15000

20000

n=100

Frequency

85 90 95 100 105 110 115 120

05000

10000

15000

20000

In the following lecture, we will revisit confidence intervals and explain one can construct them.

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