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let x_(n)sim exponential(n) . show that x_(n)arrow 0 , that is, the sequence x_(1),x_(2)- comwen, o probability to zero random variable

Question

Let X_(n)sim Exponential(n) . Show that X_(n)arrow 0 , that is, the sequence x_(1),x_(2)- comwen, o
probability to zero random variable X.
(4 marks)

Let X_(n)sim Exponential(n) . Show that X_(n)arrow 0 , that is, the sequence x_(1),x_(2)- comwen, o probability to zero random variable X. (4 marks)

Solution

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Resposta

To show that the sequence converges to 0, we need to demonstrate that for any positive value , the probability of approaches 0 as approaches infinity.Given that , we know that the expected value of is and the variance is .Now, let's consider the probability of . This can be expressed as the complement of the probability that , which is .To find the probability that , we can use the cumulative distribution function (CDF) of the exponential distribution. The CDF of an exponential random variable with rate parameter is given by .In our case, the rate parameter of is , so the CDF of is .Now, let's consider the probability that . This can be expressed as the probability that falls within the interval . Using the CDF, we can calculate this probability as .Since is a symmetric random variable around 0, we have . Therefore, .Now, let's substitute this expression into the probability that : .As approaches infinity, the term approaches 1, since the exponential distribution has a tail probability that approaches 0. Therefore, the probability that approaches 0.Hence, we have shown that for any positive value , the probability of approaches 0 as approaches infinity. This implies that the sequence converges to 0 in probability.