Probability Calculus 2020/2021, Homework 8 (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).
13. Let X1, X2, . . . be independent random variables from a distribution with density g(x) = (m + 1)xm1[0,1](x). Find the limit, in terms of almost sure convergence, of the sequence
(X1)n+1+ (X2)n+1+ . . . + (X`)n+1+ `
` + 2 , ` = 1, 2, . . . .
ANSWER:
Solution:
14. We carry out the following experiment, consisting of infinitely many independent steps: in step number ` we toss a coin, for which the probability of obtaining heads is equal to n
k + 1 2`. Let S` denote the number of heads obtained in the first ` steps. Find the limit, in terms of convergence in probability, of the sequence S`
` , ` = 1, 2, . . . .
ANSWER:
Solution:
