DISTRIBUCIONES CONTINUAS DE PROBABILIDAD

DISTRIBUCION UNIFORME

Parámetros

min <- 5
max <- 10

Generación de aleatorios uniformes:

u <- runif(10000, min, max)
head(u)

## [1] 8.104 8.714 5.493 6.509 9.461 7.360

dunif(7, min, max)

## [1] 0.2

punif(8, min, max)

## [1] 0.6

qunif(0.9, min, max)

## [1] 9.5

curve(dunif(x, 5, 10), from = min - 0.5, to = max + 0.5, lwd = 2, main = expression(f(x) == 
    frac(1, 5)))

mean(u)

## [1] 7.5

sd(u)

## [1] 1.442

hist(u, freq = FALSE, main = "Histograma de aleatorios uniformes u", ylab = "Densidad", 
    col = "blue")

DISTRIBUCION EXPONENCIAL

Parámetro

l <- 2

Generación de aleatorios

ex <- rexp(10000, l)
head(ex)

## [1] 0.15605 0.07427 0.16934 0.65097 0.17603 0.22679

dexp(1.5, l)

## [1] 0.09957

pexp(2.2, l)

## [1] 0.9877

qexp(0.8)

## [1] 1.609

curve(dexp(x, l), from = -1, to = 4, lwd = 2, col = "green", main = expression(f(x) == 
    2 * e^(-2 * x)))

mean(ex)

## [1] 0.4973

sd(ex)

## [1] 0.5035

hist(ex, freq = FALSE, main = "Histograma de ex", col = "yellow")

DISTRIBUCION NORMAL

Parámetros

media <- 2
desv <- 7
normales <- rnorm(10000, media, desv)
head(normales)

## [1]  7.4433  4.1331 -0.6274  0.9905  1.2648  2.0581

dnorm(1.5, media, desv)

## [1] 0.05685

pnorm(2.2, media, desv)

## [1] 0.5114

qnorm(0.8, media, desv)

## [1] 7.891

curve(dnorm(x, media, desv), from = media - 4 * desv, to = media + 4 * desv, 
    lwd = 2, col = "yellow", main = expression(paste(frac(1, sigma * sqrt(2 * 
        pi)), " ", plain(e)^{
        frac(-(x - mu)^2, 2 * sigma^2)
    })), cex = 1.2)

mean(normales)

## [1] 2.013

sd(normales)

## [1] 6.933

hist(normales, freq = FALSE, main = "Histograma de aleatorios normales", ylab = "Densidad", 
    col = "gray")

FUNCION GAMMA

gamma(1)

## [1] 1

gamma(0.5)

## [1] 1.772

sqrt(pi)

## [1] 1.772

gamma(11) == factorial(10)

## [1] TRUE

curve(gamma(x), from = 0.1, to = 4, col = "violet", lwd = 3, main = expression(Gamma(x) == 
    integral(t^(x - 1) * plain(e)^-t * dt, 0, infinity)))

DISTRIBUCION GAMMA

Parámetros

l <- 2
r <- 3

Generación de aleatorios

ga <- rgamma(10000, l, r)
dgamma(1.5, l, r)

## [1] 0.15

pgamma(2.2, l, r)

## [1] 0.9897

qgamma(0.5, l, r)

## [1] 0.5594

curve(dgamma(x, l, r), from = -0.5, to = 3, lwd = 3, main = expression(f(x) == 
    frac(lambda^r, Gamma(r)) * x^(r - 1) * plain(e)^-(lambda * x)))

mean(ga)

## [1] 0.67

sd(ga)

## [1] 0.4763

hist(ga, freq = FALSE, main = "Histograma de aleatorios gamma ", ylab = "Densidad", 
    col = "green")

DISTRIBUCION CHI-CUADRADO

Parámetro

v <- 3

Generación de aleatorios

ji <- rchisq(10000, l, r)
dchisq(1.5, v)

## [1] 0.2308

pchisq(2.2, v)

## [1] 0.4681

qchisq(0.01, v)

## [1] 0.1148

curve(dchisq(x, v), from = -0.5, to = 20, lwd = 3, main = expression(f(x) == 
    frac(1, 2^(frac(v, 2)) * Gamma(frac(v, 2))) * x^(frac(v, 2) - 1) * plain(e)^(-frac(x, 
        2))))

mean(ji)

## [1] 5.052

sd(ji)

## [1] 4.022

hist(ji, freq = FALSE, main = "Histograma de aleatorios chi-cuadrado", ylab = "Densidad", 
    col = "red")

DISTRIBUCION t DE STUDENT

v grados de libertad

v <- 3

Generación de aleatorios

tds <- rt(1000, v)
head(tds)

## [1] -0.5627  0.8447  0.2205 -1.0093  1.6758 -1.3754

dt(-1.5, v)

## [1] 0.12

pt(2.2, v)

## [1] 0.9424

qt(0.9, v)

## [1] 1.638

dt(1.5, v)

## [1] 0.12

pt(-2.2, v)

## [1] 0.05759

qt(0.1, v)

## [1] -1.638

curve(dt(x, v), from = -4, to = 4, lwd = 3, col = "blue", main = expression(f(x) == 
    frac(Gamma(frac(v + 1, 2)), sqrt(v * pi) * Gamma(frac(v, 2))) * (1 + frac(x^2, 
        v))^(-frac(v + 1, 2))))

mean(tds)

## [1] 0.02897

sd(tds)

## [1] 1.614

hist(tds, xlim = c(-4.5, 4.5), freq = FALSE, breaks = 50, main = "Histograma de aleatorios t de Student tds", 
    ylab = "Densidad", col = "blue")