9 Probability and Statistics

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] 9 4 3 2

Select 10 numbers from 0 to 100

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

#>  [1] 26 30 19 65 43 53 90 76 23 39

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

#>  [1] 30 68 57 14 20 21  1 87 28 55

• Select students’ grades randomly

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

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

• Tossing a fair coin 10 times

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

#>  [1] "H" "H" "H" "H" "T" "H" "H" "T" "T" "H"

• Tossing a biased coin 10 times

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

#>  [1] "T" "T" "H" "H" "H" "H" "T" "H" "H" "T"

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.

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

• 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