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Department of Electrical Engineering and Computer Sciences
EECS 126: Probability and Random Processes
Discussion 6
1. Finite Boundary Times
Consider the random walk , where the Xi are iid with mean zero and variance 1 (note that they do not have to be discrete). Show that almost surely the random walk will leave the interval [−a,a] in finite time.
Hint: Let T be the first time that the walk leaves the interval [−a,a], and show that limn→∞ P(T > n) = 0.
2. Confidence Interval Comparisons
In order to estimate the probability of a head in a coin flip, p, you flip a coin n times, where n is a positive integer, and count the number of heads, Sn. You use the estimator ˆp = Sn/n.
(a) You choose the sample size n to have a guarantee

Using Chebyshev Inequality, determine n with the following parameters. Note that you should not have p in your final answer.
(i) Compare the value of n when 1 to the value of n when
(ii) Compare the value of n when 05 to the value of n when
(b) Now, we change the scenario slightly. You know that p ∈ (0.4,0.6) and would now like to determine the smallest n such that
.
Use the CLT to find the value of n that you should use. Recall that the CLT states that the sum of IID random variables tends to a normal distribution with the sample mean and variance as it’s parameters for n large enough.
3. Characteristic Function Basics
The definition of the characteristic function for random variable X is ϕX(t) = E[eitX]. It has many important properties – most notably that there is a bijection between the CDF (and therefore also PDF) of a random variable and its characteristic function. This problem goes over some of its basic properties.
(a) Let X be a random variable that takes on the values 1 and −1 with equal probability. Show that ϕX(t) = cos(t).
(b) Let X be a uniform random variable on the interval [a,b]. Show that
.
. What happens if b = −a?
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(c) Show that ϕX(−t) = ϕX(t), where the bar means take the complex conjugate. Use this fact to argue that if the distribution of X is symmetric about the origin, then the characteristic function is strictly real.
(d) Show that
.
This can be particularly useful for computing higher moments of random variables.
(e) Show that that for independent X1,…,Xn and scalars a1,…,an,
ϕa1X1+…+anXn(t) = ϕX1(a1t) · … · ϕXn(ant).
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