Background The Hubble constant is a fundamental cosmological parameter used to describe the size and age of the universe. It relates the distance and velocity of objects relative to earth via the relationship v = nd. Astrophysicists estimate the Hubble constant , from observed ratios H = $ for a sample of galaxies. However, observations of H for any given galaxy can vary considerably due to measurement error. Though seemingly simple, measurements of velocity and distance are challenging to obtain with accuracy, and continue to be refined (see a recent article). Velocity is calculated from spectroscopic measurements, which are less complicated to obtain compared with distance measurements. For the latter, "[reliable 'distance indicators', such as variable stars and supernovae, must be found in ... a sample of galaxies that are far enough away that motions due to local gravitational influences are negligibly small" (quote from Space Science Short). The primary objective of the Hubble Space Telescope (HST), launched in 1990, is to collect data that will lead to more precise estimates of the Hubble constant. In 2001, Freedman et al published a set of measurements based on novel distance indicators and HST data; distance-velocity ratios from measurements for 36 type la supernovae using their method of calculation are shown in Figure 1. 0.0 density 0.04- 0.02 0100- 70 75 80 velocity/distance Figure 1: Histogram of # values from 36 type la supernovae, with a smooth overlay to aid in visualizing the shape of the distribution. The exact values in the sample are shown as ticks below the histogram. The sample mean and variance are H = 72.1861 and S; = 23.7858.Objective Your goal in this assignment is to estimate the age of the universe based on an estimate of the Hubble constant , and to quantify the quality of the estimate. The age of the universe can be expressed in years as 8 = where c is the conversion factor (4pc/km) (s/wear) 2 = 978, 440, 076, 094. Assumptions Assume throughout that HI, ..., Has ~ gamma(a, 8), so that the common density is: f(h; a, 8) = BOT (a) ha-le # h20 Note that this exam works with the gamma density as parameterized in the textbook - that is, with shape parameter o and scale parameter S. Assume further that the common mean is in fact the Hubble constant; denote this by 7 = E[ H.]. Consider the estimators f = H = 2.#4 and 0 = $. Problems 1. The following parts guide you through finding the distribution of 0. (a) Show that c-H has a gamma (no, # ) distribution. Write its density. (b) Show that 6 = cH-' has an inverse gamma distribution with density (cn ) no g(u; a, 8) = = 1no-le- gnal(no) For short, U ~ IG (no, " ). (Hint: for > > 0, the function = is monotone decreasing.)