1. Suppose X is a continuous, non-negative random variable with pdf f and expected value μ.

(a) Prove the Markov inequality:

P(X ≥ x) ≤ μ/x

(b) Suppose X is a continuous, non-negative random variable with pdf f and expected value μ and variance σ². Use the Markov inequality to prove the Chebyshev inequality

P(|X - μ| ≥ x) ≤ σ²/x² 

2. Let X ~ Exp(λ). Calculate the cdf and pdf of Y = |X - μ|

3. Let X ~ Exp(1/2).

(a) Calculate both sides of the Markov and Chebyshev inequalities as functions of x > 0.

(b) Use the inverse transform method to simulate 100 realisations of X and plot on the same graph (for 1 < x < 4)

(i) the crude Monte Carlo estimate of the left-hand side of the Chebyshev inequality;

(ii) the calculated left-hand side of the Chebyshev inequality; and

(iii) the calculated right-hand side of the Chebyshev inequality. Supply your plot and your code.

4. Simulate 100 coin flip experiments with a biased coin with "success" probability (chance of 1) p = 1/8. Calculate and record the observed proportion of successes (sample mean). Repeat this experiment 1000 times, and plot the empirical cdf of the 1000 sample means you observed and indicate on your plot the expected proportion of successes (true mean). Supply your plot and code.

5. Simulate 10 outcomes from the Exp(8) distribution using the inverse transform method. Calculate and record the average of these outcomes (sample mean). Re¬peat this experiment 1000 times, and plot the empirical cdf of the 1000 sample means you observed and indicate on your plot the expected average of these out¬comes (true mean). Supply your plot and code.

6. Using( the inverse transform method, simulate 10 outcomes from Y with Cauchy pdf f (y; μ, T) = T(Π(T² + (y - μ)²))^-1, with μ = 1/8 and T = 1. Calculate and record the average of these outcomes (sample mean). Repeat this experiment 1000 times, and plot the empirical cdf of the 1000 sample means you observed and indicate on your plot the mode of the Cauchy distribution p. Supply your plot and code. Comment on the shape of the empirical distribution of the sample mean. Explain any differences you observe.