Limit theorems

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Problem: Let \( \xi_{1}, \xi_{2}, \ldots \) be a sequence of independent random variables, each of which has the Poisson distribution. Prove that the series \( \sum_{n=1}^{\infty} \xi_{n} \) almost surely converges only when the series \( \sum_{n=1}^{\infty} E \xi_{n} \) converges.

15.7.1 Limit theorems

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Problem: Let \( \xi_{1}, \xi_{2}, \ldots \) is the sequence of such independent non-negative random variables that \( p\left(\xi_{n}>c\right)=0 \). Prove that the series \( \sum \xi_{n} \) almost surely converges only when the series \( \sum E \xi_{n} \) converges.

15.7.2 Limit theorems

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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

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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

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