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)
Solution
4.4
(227 Votos)
Ornela
Elite · Tutor por 8 anos
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.