require(ggplot2) # calling "ggplot2" to create graphs




## problem 1

fctrl <- function(n)  
  # making the function to calculate factorials
  # n = a non-negative integer  
  # NOT calculate other than non-negative integer 
  # out = result of calculation
  
  if(n < 0 | n != round(n)){  # Not calculate negative or non-integer numbers 
    print("n must be a non-negative intenger")
    
  }else if(n == 0){  # factorial 0 equals 1 
    return(1)
    
  }else{ # otherwise, calculate according to each number 
    result <- 1  # substitution to calculate the function
    for(i in 1:n){result = result * i  # calculate i to previous result for loop 
    }
    return(result)  #return the result
  }
  
fctrl(-5)  # ensure when n is negative
fctrl(0)   # ensure when n is 0
fctrl(10) == factorial(10)  #ensure the function




## problem 2

chs <- function(n, k) {# making the function to calculate combination
  # argument1 n = non-negative integer  
  # argument2 k = non-negative integer from 0 to n
  # return value = result

  if(n<k){ # in the case of k is greater than n, calculation is impossible
    print("k must be greater than n") # alert k must be greater than n
    
  }else{ #otherwise, calculate combination
    result <- fctrl(n) / (fctrl(k) * fctrl(n - k))
    return(result)  
  }
}
  
chs(5, 3) == choose(5, 3)  # ensure the function
chs(3, 5)  # ensure the alerting



## problem 3

pmfbnmnl <- function(x, n, p){
  # making the function to calculate binominal probability
  # argument1: n = non-negative integer  
  # argument2: x = non-negative integer which means bernoulli trials
  # argument3: p = number between 0 and 1 which means probability
  
  result <- chs(n, x) * p^x * ((1-p) ^(n-x)) # calculate binominal binominal probability
  return(result) 
}
  
pmfbnmnl(1, 10, 0.5) # ensure the function
dbinom(1, 10, 0.5)




## problem 4

temp.data.frame <- data.frame(p=NA, n=NA, x=NA, result=NA)
  # creat an empty data frame called "temp.data.frame" (temporary data)
  # in order to input arguments and make them be rewrited by final data frame

final.data.frame <- data.frame(p=NULL, n=NULL, x=NULL, result=NULL)
  # creat an empty data frame called "final.data.frame" (final data)
  # all arguments are NULL in order to combine final data frame to temprary data frame after 

for(p in c(0.5, 0.7)){         # loop for probability (p = 0.5 and p = 0.7)
  for(i in c(2, 5, 10, 50)){   # loop for n = 2, 5, 10, 50
    for(x in 0:i){             # loop for x = 0 ~ n
      # each for command are for three argments to imput temporary data frame
      temp.data.frame[1, ] <- c(paste("p =", p), paste("n =", i), x, pmfbnmnl(x, i, p))
      final.data.frame <- rbind(final.data.frame, temp.data.frame)
      # input p, n, x, and result into the temporary data frame
      # combine final.data.frame with temp.data.frame
      # final data frame now includes all results
    }
  }  
}

head(final.data.frame)
 # display first six data in final data frame 

tail(final.data.frame)
 # display last six data in final data frame
 # check how many results are included in final data frame

final.data.frame$n <- factor(final.data.frame$n, 
                             levels = c("n = 2", "n = 5", "n = 10", "n = 50"))
# factoring the column, n, to create the plots according to ascending order
# if this command lacks, the plots appear according to word information

prob4.opts <- list(geom_bar(aes(x = as.integer(x), y = as.numeric(result)), stat="identity"), 
                   facet_wrap(~n), ylab("probability"), xlab("number of success"))
 # save the format of the way to show the graphs (x and y bar, stat, and labels)

ggplot(data=final.data.frame[1:71, ]) + prob4.opts + ggtitle("case: p = 0.5")
 # output the plot of "p = 0.5" which is assigned number 1 to 71 according to n

ggplot(data=final.data.frame[72:142, ]) + prob4.opts + ggtitle("In The Case of p = 0.7")
 # output the plot of "p = 0.7" which is assigned number 72 to 142 according to n