Moulinex is a well-known brand on the market that produces blenders. The time needed to produce one blender is expressed in minutes and can be modelled as a continuous random variable X with density function . f(x) = (a (x - 53). (57-x)3 if 53 ≤ x ≤ 57, otherwise You may assume that the production times for producing different blenders are independent of each other. a. Show that a = 1 / 51.2. b. Determine the median production time and the mode of the production time of the blenders of the brand Moulinex. c. Determine the probability that the total production time needed to produce 50 blenders exceeds 45 hours. d. What is the maximum number of blenders that can be produced with a total production time of at most 40 hours with a probability of at least 95%? Answer question 3

Definition Definition Measure of central tendency that is the value that occurs most frequently in a data set. A data set may have more than one mode if multiple categories repeat an equal number of times. For example, in a data set with five item—3, 5, 5, 29, 473—the mode is 5 because it occurs twice and no other value occurs more than once. On a histogram or bar chart, the element with the highest bar represents the mode. Therefore, the mode is sometimes considered the most popular option. The mode is useful for nominal or categorical data (e.g., the most common color car that users purchase), but it is problematic for continuous data because it is more likely not to have any value that is more frequent than the other.