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

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

#> [1] 4 8 7 9

---

Select 10 numbers from 0 to 100

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

#>  [1] 73 65 57 85 51  2 72 55 80 86

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

#>  [1] 75 29 76 85 42 13 32  6 31 43

---

• Select students’ grades randomly

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

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

• Tossing a fair coin 10 times

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

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

• Tossing a biased coin 10 times

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

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

---

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 $\to$ number of Bernoulli trials

  • prob $\to$ probability of success

  • n $\to$ 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 $\to$ mean of the distribution $(\mu )$

  • sd $\to$ standard deviation of the distribution $(\sigma )$

  • n $\to$ 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 $P(45<X<55)$.

• $P(a<X\le b)=F(b)-F(a)$

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 $Z_{0.975}\approx 1.96$. 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