13. Exercise: Convergence in probability:
a) Suppose that Xn is an exponential random variable with parameter lambda = n. Does the sequence {Xn} converge in probability?
b) Suppose that Xn is an exponential random variable with parameter lambda = 1/n. Does the sequence {Xn} converge in probability?
c) Suppose that the random variable in the sequence {Xn} are independent, and that the sequence converges to some number a, in probability.
Let {Yn} be another sequence of random variables that are dependent, but where each Yn has the same distribution (CDF) as Xn. Is it necessarily true that the sequence {Yn} converges to a in probability?
a) yes
b) no
c) yes
a) Well, the exponential random variable with parameter lambda = n sounds like it's doing some serious stretching before it converges. So, no, the sequence {Xn} probably won't converge in probability. It's like trying to catch a greased pig at a rodeo. Slippery situation!
b) Ah, now we have an exponential random variable with parameter lambda = 1/n. It's getting smaller and smaller as n goes to infinity, just like my chances of winning the lottery. But hey, in this case, the sequence {Xn} might actually converge in probability. It's like trying to catch a snail on a lazy Sunday afternoon. Slow and steady wins the race!
c) Ah, the ABCs of convergence. If {Xn} converges in probability to a number a, can we always expect {Yn} to do the same? Well, here's the catch - {Yn} is a sequence of dependent random variables. It's like having a bunch of clowns in a car. You never know what they're up to!
So, it's not necessarily true that {Yn} will converge to a in probability. It's like trying to predict the next move of a group of clowns. It's all chaos and unpredictability. Keep that circus under control!
To determine whether a sequence {Xn} converges in probability, we need to understand the concept of convergence in probability first.
Convergence in Probability:
A sequence of random variables {Xn} converges in probability to a constant a if, for any positive epsilon (ε), the probability that the absolute difference between Xn and a is greater than epsilon approaches zero as n approaches infinity. This can be mathematically represented as:
lim(n→∞) P(|Xn - a| > ε) = 0
Now let's analyze each question:
a) Suppose that Xn is an exponential random variable with parameter lambda = n. Does the sequence {Xn} converge in probability?
To determine convergence of {Xn} in probability, we need to consider the behavior of the exponential distribution as lambda approaches infinity. As lambda increases, the exponential distribution becomes increasingly concentrated around zero. Therefore, as n approaches infinity, the probability that Xn deviates from zero by more than any given epsilon decreases to zero. Hence, the sequence {Xn} converges in probability to zero.
b) Suppose that Xn is an exponential random variable with parameter lambda = 1/n. Does the sequence {Xn} converge in probability?
In this case, as n approaches infinity, lambda approaches zero. The exponential distribution with lambda = 0 corresponds to a degenerate distribution where the random variable is always zero. As a result, Xn always equals zero, regardless of the value of n. Therefore, the sequence {Xn} converges in probability to zero.
c) Suppose that the random variables in the sequence {Xn} are independent and that the sequence converges to some number a in probability.
Let {Yn} be another sequence of random variables that are dependent, but where each Yn has the same distribution (CDF) as Xn. Is it necessarily true that the sequence {Yn} converges to a in probability?
Yes, it is true. If the sequence {Xn} converges to a in probability, that means, for any given epsilon, the probability of the absolute difference between Xn and a exceeding epsilon goes to zero as n approaches infinity. Since {Yn} has the same distribution as {Xn}, Yn also converges to a in probability. The dependency between the random variables in {Yn} does not affect the convergence property as long as the distributions remain the same.
Note: The above explanations assume that the random variables {Xn} and {Yn} have well-defined cumulative distribution functions (CDFs) and are continuous random variables.
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