Probability Calculus 2020/2021, Homework 5 (two problems)
Name and Surname ... Student’s number ...
In the problems below, please use the following: as k – the sum of digits in your student’s number; as m – the sum of the two largest digits in your student’s number;
and as n – the smallest digit in your student’s number plus 1. For example, if an index number is 609999: k = 42, m = 18, n = 1.
Please write down the solutions (transformations, substitutions etc.), and additio- nally provide the final answer in the space specified (the answer should be a number in decimal notation, rounded to four digits).
7. Let X be a random variable from a distribution with density g(x) = 2
n(n + 2)(1 + x)1[0,n](x).
Calculate E(kX + m).
ANSWER:
Solution:
8. An ATM dispenses bank notes of 10 PLN, 20 PLN, 50 PLN and 100 PLN. The number of 10 PLN bills withdrawn on a given day is a random variable from a Poisson distribution with parameter m, if it is Monday, Tuesday, Wednesday, Thursday or Friday and a Poisson distribution with parameter k/5, if it is Saturday or Sunday. We count the number of days between December 1st and n + 21-st of 2020 (inclusive), when no 10 PLN bills are withdrawn. How many such days, on average, will there be during this period?
ANSWER:
Solution:
