For , let be a Poisson random variable with parameter . Compute
e^(-1/n)
correct
What can you conclude?
in probability, but does not converge in probability
in probability, in probability, and converges.
and in probability, but does not converge.
incorrect
? STANDARD NOTATION
To compute , let's first recall the probability mass function of a Poisson random variable. For a Poisson random variable with parameter , the probability mass function is given by:
where is the number of occurrences, and is the parameter of the Poisson distribution.
In this case, we are given that is a Poisson random variable with parameter . Substituting into the probability mass function, we have:
Now, we need to compute .
To do this, we'll plug in for in the previous equation:
We can simplify this expression further using the properties of exponents:
Finally, we can conclude that is the answer. This means that, in probability, converges to 0 as approaches infinity.