(1/4)*(12/51)
choose(52,2)
choose(13,2)/choose(52,2)
13/52*12/51
13*12(52*51)

(13*12/(52*51))

  #Prawdopodobieństwo całkowite

PX1 = 0.1; PX2 = 0.4 ; PX3 = 0.5
PApwX1 = 0.05 ; PApwX2 = 0.03 ; PApwX3 = 0.03
(PA <- PX1*PApwX1+PX2*PApwX2+PX3*PApwX3)


  #wzor bayessa

PX1 = 0.1; PX2 = 0.4 ; PX3 = 0.5
PApwX1 = 0.05 ; PApwX2 = 0.03 ; PApwX3 = 0.03
PA <- PX1*PApwX1+PX2*PApwX2+PX3*PApwX3
(PX3pwA <- (PX3*PApwX3)/PA)


#Rozkłady (2)
#-rozkład Bernouliego: Zobacz :Bernouli
#generator (rbinom),
#gęstość (dbinom),
#dystrybuanta (pbinom),
#kwantyle (qbinom)

dbinom(1, size=9, prob=0.02)

dbinom(2, size=10, prob=0.02)

#Zadanie 4 - rozwiązanie (a)
#Oznaczenia
#PS(sukces)=0.98
#PP(porażka)=0.02

PS = 0.98; PP = 0.02
(P = choose(9,1)*PS^8*PP^1)

# lub
(P = dbinom(1, size=9, prob=0.02))



#4b

PS = 0.98; PP = 0.02
(P = choose(10,2)*PS^8*PP^2)

# lub
(P = dbinom(2, size=10, prob=0.02))


dpois(6, lambda=4.1)


ppois(6, lambda=4.1)                # lower tail

ppois(6, lambda=4.1, lower=FALSE)   # upper tail


n = 410;q=0.01; lambda = q*n
(dpois(6, lambda=q*n))


# lub
(lambda^6*exp(-lambda)/factorial(6))


lambda<-2
x<-rpois(200,lambda)
hist(x,breaks = 15, probability = T,col = "lightblue")

dpois(3, lambda=3)



# rozklad jednostajny
x<-runif(5,min = 2, max = 5)
dunif(x, min = 2, max = 5)


punif(x, min = 2, max = 5) #prawdopodobeństwo że los będzie mniejszy od x


min = 2 ; max = 14; czas = 6
# będzie
(p <- punif(czas,min = 2,max = 14,lower.tail = T))

# prawdopodobienstwo że x > 84 dla m = 96 oraz sigma = 6
pnorm(84, mean=96, sd=6, lower.tail=FALSE)

z<-rnorm(200,mean = 5,sd=2)
hist(z,breaks = 15, probability = T,col = "lightblue")
lines(density(z),col="red", lwd = 2)


sr = 96 ; sd = 6; kwota = 84
# prawdopodobieństwo że x < kwota
pnorm(kwota,mean = 96,sd = 6)


#prawdopodobieństwo że x > 3 przy lambda = 3
pexp(3, rate=1/3, lower.tail=FALSE)

# rozklad t studenta
?pt
x<-seq(-5,2,by =.2)
x1 <- rt(200, df-1)
x2 <- rt(200, df-2)
x3 <- rt(200, df-3)
x5 <- rt(200, df-5)
x10 <- rt(200, df-10)

hist((x1), breaks=15, col="red", probability = T)
lines(density(x1), col="blue", lwd=2)
plot(dt(x,df=2),type="l",col="red",ylim=c(0,.5))
lines(dt(x,df=5),type="l",col="blue")
lines(dt(x,df=10),type="l",col="green")
lines(dnorm(x,0,2),type="l",col="grey")


#rozklad chi-kwadrat
y<- sum(dnorm(x,0,2)2-dnorm)