Verify the identity. tan²(u) − sin²(u) = tan²(u) sin²(u) tan²(u) − sin²(u) = sin²(u) − sin²(u) cos²(u) cos²(u) = sin²(u) − cos²(u) = 1 − cos²(u) cos²(u) = sin²(u) cos²(u) · 1 − = tan²(u).
a. Derive the cdf of Yn and verify the convergence to an Exponential distribution with rate parameter 1/θ.
b. Calculate the expected value and variance of Yn and compare them to the expected value and variance of an Exponential distribution with rate parameter 1/θ.
c. Simulate Yn for various values of n and compare the empirical distribution to the theoretical Exponential distribution with rate parameter 1/θ.
d. Investigate the behavior of Yn as θ changes, and determine the impact on the convergence to an Exponential distribution.