9 Probability and Statistics

(AST230) R for Data Science

Generating random data

• The function sample() is used to generate random values from a vector, and it has the following arguments:

  • x \rightarrow A vector of outcome you want to sample from
  • size \rightarrow The number of samples (observations) you want to draw
  • replace \rightarrow It can take either TRUE or FALSE
  • prob \rightarrow Specifies probability of selection of different elements of x

sample(x = 1:10, size = 4, replace = F)

[1]  7  4 10  2

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Select 10 numbers from 0 to 100

sample(x = 0:100, size = 10, replace = F) # replace=FALSE

 [1] 76 23  9 93 19 31 15 38 65 11

sample(x = 0:100, size = 10, replace = T) # replace=TRUE

 [1]  31  47  79  27   1  52  48  91  57 100

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• Select students’ grades randomly

sample(replace = TRUE, x = LETTERS[1:4], size = 10)

 [1] "B" "D" "A" "D" "C" "A" "A" "D" "D" "D"

• Tossing a fair coin 10 times

sample(replace = TRUE, x = c("H", "T"), size = 10)

 [1] "H" "H" "H" "H" "H" "T" "H" "H" "T" "T"

• Tossing a biased coin 10 times

sample(replace = TRUE, x = c("H", "T"), size = 10, prob = c(.7, .25))

 [1] "H" "H" "H" "H" "H" "T" "H" "H" "H" "H"

Use of initial seed in generating random numbers

Without seed:

# No seed
sample(1:10, 3)

[1] 1 2 5

# No seed
sample(1:10, 3)

[1] 4 5 7

# No seed
sample(1:10, 3)

[1] 4 1 7

With seed:

set.seed(100)
sample(1:10, 3)

[1] 10  7  6

set.seed(100)
sample(1:10, 3)

[1] 10  7  6

set.seed(100)
sample(1:10, 3)

[1] 10  7  6

Useful functions related to probability distributions:

• R has built in many functions for conveniently working with a large number of distributions.

• The quantities that are of main interest from any probability distribution are:

  • Probability density function (pdf for continuous variable) or probability mass function (pmf for discrete variable)
  • Cumulative distribution function (cdf)
  • Quantile function (inverse cdf)
  • Generating random sample from respective distributions.

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Distribution	Density Function	Cumulative Distribution	Quantile	Random Variates
Normal	dnorm()	pnorm()	qnorm()	rnorm()
Poisson	dpois()	ppois()	qpois()	rpois()
Binomial	dbinom()	pbinom()	qbinom()	rbinom()
Uniform	dunif()	punif()	qunif()	runif()

• Such functions are available for other probability distributions, such as exponential, logistic, Chi-squared etc.

rbinom() and rnorm

• rbinom() is used to draw a sample from a binomial distribution

  • size \rightarrow number of Bernoulli trials

  • prob \rightarrow probability of success

  • n \rightarrow number of observations

• Draw a sample of size 8 from B(10, 0.75)

rbinom(size = 10, prob = .75, n = 8)

[1] 9 8 8 6 8 7 9 7

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• rnorm() is used to draw a sample from a normal distribution

  • mean \rightarrow mean of the distribution (\mu)

  • sd \rightarrow standard deviation of the distribution (\sigma)

  • n \rightarrow number of observations

• Draw a sample of size 5 from N(10, 16)

rnorm(mean = 10, sd = 4, n = 5)

[1] 14.743527  8.970948 11.748854  8.539669 11.986696

pnorm()

• For X \sim N(50, 3^2), find P(45<X<55).

• P(a < X ≤ b) = F(b) − F(a)

pnorm(q = 55, mean = 50, sd = 3) - 
  pnorm(q = 45, mean = 50, sd = 3) 

[1] 0.9044193

dnorm()

• For X \sim Bin(10, 0.5), find P(X=5).

dbinom(x = 5, size = 10, prob = 0.5)

[1] 0.2460938

qnorm()

• Let Z follows a standard normal distribution. Then the 0.975−quantile is Z_{0.975} ≈ 1.96. It means the probability of sampling a value less than or equal to 1.96 is 0.975 or 97.5%

qnorm(p = 0.975, mean = 0, sd = 1)

[1] 1.959964