Probability and distribution with r

• We outline the basic ideas of probability and the functions that Rhas for random sampling and handling of theoretical distributions... • We outline the basic ideas of probability and t

Introduction to

Probability and Distributions with R

Nguyen An Khuong, HCMUT, VNU-HCM

Ngày 15 tháng 9 năm 2016

Probability and distributions with R

Randomness

Motivations

• Gambling

• Real life problems

• Computer Science: cryptology, coding theory, algorithmiccomplexity,

Probability and distributions with R

Randomness

Motivations

• Gambling

• Real life problems

• Computer Science: cryptology, coding theory, algorithmiccomplexity,

Probability and distributions with R

Randomness

Motivations

• Gambling

• Real life problems

• Computer Science: cryptology, coding theory, algorithmiccomplexity,

Probability and distributions with R

Randomness

Motivations

• Gambling

• Real life problems

• Computer Science: cryptology, coding theory, algorithmiccomplexity,

Probability and distributions with R

Randomness

Motivations

• Gambling

• Real life problems

• Computer Science: cryptology, coding theory, algorithmiccomplexity,

Probability and distributions with R

Randomness

Motivations

• Gambling

• Real life problems

• Computer Science: cryptology, coding theory, algorithmiccomplexity,

Probability and distributions with R

Randomness

Motivations

• Gambling

• Real life problems

• Computer Science: cryptology, coding theory, algorithmiccomplexity,

Motivations

• Gambling

• Real life problems

• Computer Science: cryptology, coding theory, algorithmic

complexity,

Probability and distributions with R

Randomness

Randomness

Which of these arerandom phenomena?

• The number you receive when rolling a fairdice

• The sequence for lottery special prize (by law!)

• Your blood type (No!)

• You met the red light on the way to school

• The traffic light isnotrandom It has timer

• The pattern ofyour ridingis random

So what is special about randomness?

In thelong run, they are predictable and haverelative frequency(fraction

of times that the event occurs over and over and over)

Probability and distributions with R

Randomness

Randomness

Which of these arerandom phenomena?

• The number you receive when rolling a fairdice

• The sequence for lottery special prize (by law!)

• Your blood type (No!)

• You met the red light on the way to school

• The traffic light isnotrandom It has timer

• The pattern ofyour ridingis random

So what is special about randomness?

In thelong run, they are predictable and haverelative frequency(fraction

of times that the event occurs over and over and over)

Probability and distributions with R

Randomness

Randomness

Which of these arerandom phenomena?

• The number you receive when rolling a fairdice

• The sequence for lottery special prize (by law!)

• Your blood type (No!)

• You met the red light on the way to school

• The traffic light isnotrandom It has timer

• The pattern ofyour ridingis random

So what is special about randomness?

In thelong run, they are predictable and haverelative frequency(fraction

of times that the event occurs over and over and over)

Probability and distributions with R

Randomness

Randomness

Which of these arerandom phenomena?

• The number you receive when rolling a fairdice

• The sequence for lottery special prize (by law!)

• Your blood type (No!)

• You met the red light on the way to school

• The traffic light isnotrandom It has timer

• The pattern ofyour ridingis random

So what is special about randomness?

In thelong run, they are predictable and haverelative frequency(fraction

of times that the event occurs over and over and over)

Probability and distributions with R

Randomness

Randomness

Which of these arerandom phenomena?

• The number you receive when rolling a fairdice

• The sequence for lottery special prize (by law!)

• Your blood type (No!)

• You met the red light on the way to school

• The traffic light isnotrandom It has timer

• The pattern ofyour ridingis random

So what is special about randomness?

In thelong run, they are predictable and haverelative frequency(fraction

of times that the event occurs over and over and over)

Probability and distributions with R

Randomness

Randomness

Which of these arerandom phenomena?

• The number you receive when rolling a fairdice

• The sequence for lottery special prize (by law!)

• Your blood type (No!)

• You met the red light on the way to school

• The traffic light isnotrandom It has timer

• The pattern ofyour ridingis random

So what is special about randomness?

In thelong run, they are predictable and haverelative frequency(fraction

of times that the event occurs over and over and over)

Probability and distributions with R

Randomness

Randomness

Which of these arerandom phenomena?

• The number you receive when rolling a fairdice

• The sequence for lottery special prize (by law!)

• Your blood type (No!)

• You met the red light on the way to school

• The traffic light isnotrandom It has timer

• The pattern ofyour ridingis random

So what is special about randomness?

In thelong run, they are predictable and haverelative frequency(fraction

of times that the event occurs over and over and over)

Probability and distributions with R

Randomness

Randomness

Which of these arerandom phenomena?

• The number you receive when rolling a fairdice

• The sequence for lottery special prize (by law!)

• Your blood type (No!)

• You met the red light on the way to school

• The traffic light isnotrandom It has timer

• The pattern ofyour ridingis random

So what is special about randomness?

In thelong run, they are predictable and haverelative frequency(fraction

of times that the event occurs over and over and over)

Randomness

Which of these arerandom phenomena?

• The number you receive when rolling a fairdice

• The sequence for lottery special prize (by law!)

• Your blood type (No!)

• You met the red light on the way to school

• The traffic light isnotrandom It has timer

• The pattern ofyour ridingis random

So what is special about randomness?

In thelong run, they are predictable and haverelative frequency(fraction

of times that the event occurs over and over and over)

Probability and distributions with R

Randomness

Randomness in Statistics

• Randomness and probability: central to statistics

• Empirical fact: Most experiments and investigations are not perfectlyreproducible

• The degree of irreproducibility may vary:

• Some experiments in physics may yield data that are accurate tomany decimal places,

• whereas data on biological systems are typically much less reliable

• View of data as something coming from a statistical distribution:

vital to understanding statistical methods

• We outline the basic ideas of probability and the functions that Rhas for random sampling and handling of theoretical distributions

Probability and distributions with R

Randomness

Randomness in Statistics

• Randomness and probability: central to statistics

• Empirical fact: Most experiments and investigations are not perfectly

reproducible

• The degree of irreproducibility may vary:

• Some experiments in physics may yield data that are accurate tomany decimal places,

• whereas data on biological systems are typically much less reliable

• View of data as something coming from a statistical distribution:

vital to understanding statistical methods

• We outline the basic ideas of probability and the functions that Rhas for random sampling and handling of theoretical distributions

Probability and distributions with R

Randomness

Randomness in Statistics

• Randomness and probability: central to statistics

• Empirical fact: Most experiments and investigations are not perfectly

reproducible

• The degree of irreproducibility may vary:

• Some experiments in physics may yield data that are accurate tomany decimal places,

• whereas data on biological systems are typically much less reliable

• View of data as something coming from a statistical distribution:vital to understanding statistical methods

• We outline the basic ideas of probability and the functions that Rhas for random sampling and handling of theoretical distributions

Probability and distributions with R

Randomness

Randomness in Statistics

• Randomness and probability: central to statistics

• Empirical fact: Most experiments and investigations are not perfectly

reproducible

• The degree of irreproducibility may vary:

• Some experiments in physics may yield data that are accurate to

many decimal places,

• whereas data on biological systems are typically much less reliable

• View of data as something coming from a statistical distribution:vital to understanding statistical methods

• We outline the basic ideas of probability and the functions that Rhas for random sampling and handling of theoretical distributions

Probability and distributions with R

Randomness

Randomness in Statistics

• Randomness and probability: central to statistics

• Empirical fact: Most experiments and investigations are not perfectly

reproducible

• The degree of irreproducibility may vary:

• Some experiments in physics may yield data that are accurate to

many decimal places,

• whereas data on biological systems are typically much less reliable

• View of data as something coming from a statistical distribution:vital to understanding statistical methods

• We outline the basic ideas of probability and the functions that Rhas for random sampling and handling of theoretical distributions

Probability and distributions with R

Randomness

Randomness in Statistics

• Randomness and probability: central to statistics

• Empirical fact: Most experiments and investigations are not perfectly

reproducible

• The degree of irreproducibility may vary:

• Some experiments in physics may yield data that are accurate to

many decimal places,

• whereas data on biological systems are typically much less reliable

• View of data as something coming from a statistical distribution:

vital to understanding statistical methods

• We outline the basic ideas of probability and the functions that Rhas for random sampling and handling of theoretical distributions

Randomness in Statistics

• Randomness and probability: central to statistics

• Empirical fact: Most experiments and investigations are not perfectlyreproducible

• The degree of irreproducibility may vary:

• Some experiments in physics may yield data that are accurate to

many decimal places,

• whereas data on biological systems are typically much less reliable

• View of data as something coming from a statistical distribution:

vital to understanding statistical methods

• We outline the basic ideas of probability and the functions that R

has for random sampling and handling of theoretical distributions

Probability and distributions with R

Sampling with R

Random Numbers with R

• Much of the earliest work in probability theory was about games and

gambling issues, based on symmetry considerations

• The basic notion then is that of a random sample:

dealing from awell-shuffled pack of cards or picking numbered balls from awell-stirred urn

• In R, we can simulate these situations with the sample function

• If we want to pick five numbers at random from the set 1 : 40, thenyou can write

> sample(1:40,5)[1] 4 30 28 40 13

Probability and distributions with R

Sampling with R

Random Numbers with R

• Much of the earliest work in probability theory was about games and

gambling issues, based on symmetry considerations

• The basic notion then is that of a random sample:

dealing from awell-shuffled pack of cards or picking numbered balls from awell-stirred urn

• In R, we can simulate these situations with the sample function

• If we want to pick five numbers at random from the set 1 : 40, thenyou can write

> sample(1:40,5)[1] 4 30 28 40 13

Probability and distributions with R

Sampling with R

Random Numbers with R

• Much of the earliest work in probability theory was about games and

gambling issues, based on symmetry considerations

• The basic notion then is that of a random sample:

dealing from awell-shuffled pack of cards or picking numbered balls from awell-stirred urn

• In R, we can simulate these situations with the sample function

• If we want to pick five numbers at random from the set 1 : 40, thenyou can write

> sample(1:40,5)[1] 4 30 28 40 13

Probability and distributions with R

Sampling with R

Random Numbers with R

• Much of the earliest work in probability theory was about games and

gambling issues, based on symmetry considerations

• The basic notion then is that of a random sample: dealing from a

well-shuffled pack of cards or

picking numbered balls from awell-stirred urn

• In R, we can simulate these situations with the sample function

• If we want to pick five numbers at random from the set 1 : 40, thenyou can write

> sample(1:40,5)[1] 4 30 28 40 13

Probability and distributions with R

Sampling with R

Random Numbers with R

• Much of the earliest work in probability theory was about games and

gambling issues, based on symmetry considerations

• The basic notion then is that of a random sample: dealing from a

well-shuffled pack of cards or picking numbered balls from a

well-stirred urn

• In R, we can simulate these situations with the sample function

• If we want to pick five numbers at random from the set 1 : 40, thenyou can write

> sample(1:40,5)[1] 4 30 28 40 13

Probability and distributions with R

Sampling with R

Random Numbers with R

• Much of the earliest work in probability theory was about games and

gambling issues, based on symmetry considerations

• The basic notion then is that of a random sample: dealing from a

well-shuffled pack of cards or picking numbered balls from a

well-stirred urn

• In R, we can simulate these situations with the sample function

• If we want to pick five numbers at random from the set 1 : 40, then

you can write

> sample(1:40,5)[1] 4 30 28 40 13

Probability and distributions with R

Sampling with R

Random Numbers with R

• Much of the earliest work in probability theory was about games and

gambling issues, based on symmetry considerations

• The basic notion then is that of a random sample: dealing from a

well-shuffled pack of cards or picking numbered balls from a

well-stirred urn

• In R, we can simulate these situations with the sample function

• If we want to pick five numbers at random from the set 1 : 40, then

you can write

> sample(1:40,5)

[1] 4 30 28 40 13

Sampling with R

Random Numbers with R

• Much of the earliest work in probability theory was about games andgambling issues, based on symmetry considerations

• The basic notion then is that of a random sample: dealing from a

well-shuffled pack of cards or picking numbered balls from a

well-stirred urn

• In R, we can simulate these situations with the sample function

• If we want to pick five numbers at random from the set 1 : 40, thenyou can write

> sample(1:40,5)

[1] 4 30 28 40 13

Probability and distributions with R

Sampling with R

Sample function

• The first argument (x) is a vector of values to be sampled

• The second (size) is the sample size

• Actually, sample(40, 5) would suffice

since a single number isinterpreted to represent the length of a sequence of integers

• Notice that the default behavior of sample is sampling withoutreplacement

• That is, the samples will not contain the same number twice, andsize obviously cannot be bigger than the length of the vector to besampled

• If we want sampling with replacement, then we need to add theargument replace = TRUE

Probability and distributions with R

Sampling with R

Sample function

• The first argument (x) is a vector of values to be sampled

• The second (size) is the sample size

• Actually, sample(40, 5) would suffice

since a single number isinterpreted to represent the length of a sequence of integers

• Notice that the default behavior of sample is sampling withoutreplacement

• That is, the samples will not contain the same number twice, andsize obviously cannot be bigger than the length of the vector to besampled

• If we want sampling with replacement, then we need to add theargument replace = TRUE

Probability and distributions with R

Sampling with R

Sample function

• The first argument (x) is a vector of values to be sampled

• The second (size) is the sample size

• Actually, sample(40, 5) would suffice

since a single number isinterpreted to represent the length of a sequence of integers

• Notice that the default behavior of sample is sampling withoutreplacement

• That is, the samples will not contain the same number twice, andsize obviously cannot be bigger than the length of the vector to besampled

• If we want sampling with replacement, then we need to add theargument replace = TRUE

Probability and distributions with R

Sampling with R

Sample function

• The first argument (x) is a vector of values to be sampled

• The second (size) is the sample size

• Actually, sample(40, 5) would suffice

since a single number isinterpreted to represent the length of a sequence of integers

• Notice that the default behavior of sample is sampling withoutreplacement

• That is, the samples will not contain the same number twice, andsize obviously cannot be bigger than the length of the vector to besampled

• If we want sampling with replacement, then we need to add theargument replace = TRUE

Probability and distributions with R

Sampling with R

Sample function

• The first argument (x) is a vector of values to be sampled

• The second (size) is the sample size

• Actually, sample(40, 5) would suffice since a single number is

interpreted to represent the length of a sequence of integers

• Notice that the default behavior of sample is sampling withoutreplacement

• That is, the samples will not contain the same number twice, andsize obviously cannot be bigger than the length of the vector to besampled

• If we want sampling with replacement, then we need to add theargument replace = TRUE

Probability and distributions with R

Sampling with R

Sample function

• The first argument (x) is a vector of values to be sampled

• The second (size) is the sample size

• Actually, sample(40, 5) would suffice since a single number is

interpreted to represent the length of a sequence of integers

• Notice that the default behavior of sample is sampling without

replacement

• That is, the samples will not contain the same number twice, andsize obviously cannot be bigger than the length of the vector to besampled

• If we want sampling with replacement, then we need to add theargument replace = TRUE