Description

CS 70 Discrete Mathematics and Probability Theory

1 First Exponential to Die
Let X and Y be Exponential(λ1) and Exponential(λ2) respectively, independent. What is
,
the probability that the first of the two to die is X?
2 Chebyshev’s Inequality vs. Central Limit Theorem
Let n be a positive integer. Let X1,X2,…,Xn be i.i.d. random variables with the following distribution:
; ; .
(a) Calculate the expectations and variances of X1, , and
Zn .
(b) Use Chebyshev’s Inequality to find an upper bound b for P[|Zn| ≥ 2].
(c) Can you use b to bound P[Zn ≥ 2] and P[Zn ≤ −2]?
(d) As n → ∞, what is the distribution of Zn?
(e) We know that if Z ∼ N (0,1), then P[|Z| ≤ 2] = Φ(2)−Φ(−2) ≈ 0.9545. As n → ∞, can you provide approximations for P[Zn ≥ 2] and P[Zn ≤ −2]?
3 Why Is It Gaussian?
Let X be a normally distributed random variable with mean µ and variance σ2. Let Y = aX +b, where a > 0 and b are non-zero real numbers. Show explicitly that Y is normally distributed with
mean aµ +b and variance a2σ2. The PDF for the Gaussian Distribution is . One approach is to start with the cumulative distribution function ofY and use it to derive the probability density function of Y.
[1.You can use without proof that the pdf for any gaussian with mean and sd is given by the formula
where µ is the mean value for X and σ2 is the variance. 2. The drivative of CDF
gives PDF.]