1. (Distributions with Random Parameters) Suppose that the density X of red blood corpuscles in humans follows a Poisson distribution whose parameter depends on the observed individual. This means that for Jason we have X ~ Poi(mj), where mj is Jason's parameter value, while for Alice we have X ~Poi(mA), where mA is Alice's parameter value. For a person selected at random we may consider the parameter value M as a random variable such that, given that M, we have X~Poi(m); namely, Thus, if we know that Alice was chosen, then P(X k| MA)for k0,1,2,., as before Now let us assume that MExp(1), i.e., M has an exponential distribution. By (the continuous version of) the law of total probability, we obtain, for k = 0, 1, 2, . . . , 0 -エ (a) (10 points) What is the distribution that X follows. (b) (10 points) Determine the distribution of X if M has an Exp(a)-distribution 1. (Distributions with Random Parameters) Suppose that the density X of red blood corpuscles in humans follows a Poisson distribution whose parameter depends on the observed individual. This means that for Jason we have X ~ Poi(mj), where mj is Jason's parameter value, while for Alice we have X ~Poi(mA), where mA is Alice's parameter value. For a person selected at random we may consider the parameter value M as a random variable such that, given that M, we have X~Poi(m); namely, Thus, if we know that Alice was chosen, then P(X k| MA)for k0,1,2,., as before Now let us assume that MExp(1), i.e., M has an exponential distribution. By (the continuous version of) the law of total probability, we obtain, for k = 0, 1, 2, . . . , 0 -エ (a) (10 points) What is the distribution that X follows. (b) (10 points) Determine the distribution of X if M has an Exp(a)-distribution