Limit theorems

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ξ_{1}, ξ_{2}, \dots

be a sequence of independent random variables, each of which has the Poisson distribution. Prove that the series

\sum_{n = 1}^{\infty} ξ_{n}

almost surely converges only when the series

\sum_{n = 1}^{\infty} E ξ_{n}

15.7.1 Limit theorems

5.14 $

ξ_{1}, ξ_{2}, \dots

is the sequence of such independent non-negative random variables that

p (ξ_{n} > c) = 0

. Prove that the series

\sum ξ_{n}

almost surely converges only when the series

\sum E ξ_{n}

15.7.2 Limit theorems

5.14 $

Problem: What is the probability that the sum of

n

independent random variables, uniformly distributed in [0,1], will not exceed 1.

15.7.3 Limit theorems

3.08 $

Problem: The probability of sensor failure within a month is

0, 15.2000

sensors are inspected once in a month. How many spare sensors must be there, so that all sensors, having defects, could be replaced with the probability of at least 0,98 .

15.7.4 Limit theorems

2.57 $