I am confused on this one

Problem 2. (25 points) A construction company is using large machinery such as bulldozers in a construction job. Such machinery frequently breakdown and require repair. The times between successive breakdowns, T, is uncertain and is modeled as a random variable. Assuming breakdowns constitute Poisson events, T has the exponential distribution: f, (t)=/1exp(itt), r>0 where A is the mean rat of breakdowns, which depends on the conditions of the job. From previous experience under similar conditions, it is estimated that [1 has a mean of three breakdowns per month and a standard deviation of one breakdown per month. (a) Choose an appropriate prior distribution for A from the table of conjugate pairs and determine its parameters (b) Suppose the 6th breakdown is encountered 45 days after starting the job. (Assume there are 30 days in a month.) Determine the posterior distribution of ,1 and its mean and standard deviation. (c) Determine the predictive PDF and GDP of T, denoted f} (t) and FT\") , respectively, which incorporate the uncertainty in 1. (d) Determine the probabilities of the following events: (1) the time between two successive breakdowns is six days or less; (2) the time between successive breakdowns is more than one month. Compare the estimates of these probabilities based on the predictive distribution 15,. (r) and based on point estimator of .1 equal to its posterior mean