Q1. Many other probability inequalities exist besides the Markov and Chebyshev's inequalities we saw in Chapter 4. For instance:
Gauss Inequality:
For a unimodal random variable X with mode ν and τ2 = E(X - ν)2, we have:
![1268_Figure.png](https://secure.tutorsglobe.com/CMSImages/1268_Figure.png)
Vysochanskii-Petunin Inequality:
For a unimodal random variable X, for any arbitrary point α with ξ2 = E(X - α)2, we have:
![44_Figure1.png](https://secure.tutorsglobe.com/CMSImages/44_Figure1.png)
Compare (i.e. are they tighter/conservative/etc) the bounds obtained from Gauss, Vysochanskii-Petunin and Chebyshev's inequalities for:
(a) a random variable X following the uniform distribution in [0, 1]
(b) a random variable X following N(μ, 1).
Q2. If Z is a standard normal random variable, then
P(|Z| ≥ t) ≤ √(2/π) e-t^2/2/t, for all t > 0.
(a) Prove the above inequality and its companion
P(|Z| ≥ t) ≥ √(2/π) (t/1+t2)e-t^2/2.
Hint: start with working out the P(Z ≥ t).
(b) Compare the above inequality with Chebychev's inequality. You can illustrate the comparison giving specific examples by setting for instance t = 2 or any t of your choice.
Q3. (a) Let X1, . . . , Xn be independent and identically distributed random variables with mean μx and variance σ2X. Similarly let Y1, . . . , Yn be independent and identically distributed random variables with mean μY and variance σ2Y. Show that the distribution of the random varaible
Wn = (X- - Y-) - (μX - μY)/√((σ2X+ σ2Y)/n)
converges to a standard normal distribution as n → ∞.
(b) The social media usage of two UCD students, Walter and Jesse, is to be compared. Walter's usage (in hours) is recorded for nw = 50 randomly selected days and Jesse's usage is recorded for nj = 60 randomly selected days. Assume that σ2W = 0.5 and σ2J = 0.3. Find the probability that the difference in the sample means will be within 0.1 hours of the difference between the population means.
Hint: The result shown in part (a) also holds when the sample sizes are unequal:
((X- - Y-) - (μX - μY))/√(σ2X/nX + σ2Y/nY) ∼ N(0, 1) as nX, nY → ∞
(c) If nW = nJ = n, find the smallest sample size n which will allow the difference between the sample means to be within 0.1 hours of the difference between the population means with probability greater than 0.9.
Q4. Let X1, . . . , Xn be iid random variables with probability density function
fX(x) = θxθ-1, 0 ≤ x ≤ 1 0 < θ < ∞
(a) Find the method of moments estimator for θ.
(b) Find the maximum likelihood estimator for θ.