Distributions and CLT

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• __d__ functions: probability density function* or probability mass function

dnorm(0)

## [1] 0.3989423

dnorm(1)

## [1] 0.2419707

dnorm(-1)

## [1] 0.2419707

dnorm(3)

## [1] 0.004431848

x <- seq(-3, 3, by = .01)
#x
#dnorm(x)
plot(x, dnorm(x), type = "line")

## Warning in plot.xy(xy, type, ...): plot type 'line' will be truncated to first
## character

• __p*__ – cumulative distribution functions

pnorm(0)

## [1] 0.5

pnorm(-1)

## [1] 0.1586553

pnorm(2)

## [1] 0.9772499

plot(x, pnorm(x), type = "line")

## Warning in plot.xy(xy, type, ...): plot type 'line' will be truncated to first
## character

• __q*__ – quantile function

• __r*__ – random numbers

set.seed(42)
rnorm(1)

## [1] 1.370958

rnorm(10)

##  [1] -0.56469817  0.36312841  0.63286260  0.40426832 -0.10612452  1.51152200
##  [7] -0.09465904  2.01842371 -0.06271410  1.30486965

iq <- rnorm(10, mean = 100, sd = 15)
hist(iq)

many_means <- replicate(100000, mean(rnorm(100, mean = 100, sd = 15)))
hist(many_means)

hist(rlnorm(100))

many_log_means <- replicate(100000, mean(rlnorm(100)))

hist(many_log_means)