The weekly demand for fuel (in 1000s of gallons) from a particular facility is a random variable, X with pdf: f ( x ) = 12 (1 - x2 ) 1 S x52 otherwise a) [4 pts] Use R to make a plot of the probability distribution. Have your x-axis show the plot from a value of 0 to 3. Provide your code and figure. You may find the following functions useful: i. To create a vector with a provided increment: vector=seq(start_value, end_value, increment) ii. To pre-allocate a double vector with zeros: zero_vector=numeric (number_of_elements ) ifi. The syntax for a for loop is: count= 0 for(val in vector) { #Insert relevant statement count=count+1 where val is the value of the element in the vector. iv . The syntax for an if/else loop: if (test_expression1) { #Insert relevant statement } else if (test expression2) { #Insert relevant statement } else { #Insert relevant statement b) [3 pts] Show that this is a valid probability distribution. c) [8 pts] What is the cumulative distribution function for this problem? Plot this distribution using R . d) [2 pts] What is the probability of needing more than 1.5 thousand gallons? e) [2 pts] What is the probability of needing less than 1 thousand gallons? f) [2 pts] What is the probability of needing between 1.2 and 1.6 thousand gallons? 8) [3 pts] What is the expected value of X? h) [3 pts] What is the variance of X? i) [3 pts] If two thousand gallons are in stock at the beginning of the week and no new supply is due during the week, how much of the two thousand gallons is expected to be left at the end of the week