0.08- 0.06- (b) Show that UH- has an inverse gamma distribution with density = g(u) = (n)a F(a) -na-1 e 0.04- 0.02- 0.00- L 65 70 75 velocity/distance 80 85 (Hint: for x > 0, the function is monotone decreasing.) (c) Assume that H = a = EH. Express in terms of a and . What is bias()? (Note: you do not need to calculate the mean of an inverse gamma random variable look up the mean in a distribution table or on Wikipedia.) Figure 1: Histogram of H values from 36 type la supernovae, with a smooth overlay to aid in visualizing the shape of the distribution. The sample mean and variance are H = 72.1861 and S=23.7858. Let H denote the true Hubble constant. The age of the universe can be expressed in years as (d) Is consistent for ? = H where c is the conversion factor (Mpc/km) (s/year) 978, 440,076, 094. 1. Consider = CH. Compute an estimate of using the estimator and the supernova data. 2. Now suppose that H,..., H36 gamma(a, b) with density f(h) = (a) -ha-1e-Bh h>0 (a) What is the distribution of H? (Hint: refer to HW2 problem 3b.)