2. This problem copes with a new distribution: exponential distribution. Let X {x(e) R}/ be...

 

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2. This problem copes with a new distribution: exponential distribution. Let X {x(e) € R}/₁ be an i.i.d. sample of N univariate instances drawn from an exponential density function: P(x(0) | 0) = { 0 exp(-0z(0) _r(0) 20 0 x(e) < 0, where > 0 is called the rate parameter of the distribution. (a) Find a point estimate of using maximum likelihood estimation. Show all the steps in your derivation. (b) Suppose is now treated as a random variable (instead of a fixed parameter as above) with prior density p(0) of the following form: p(0) x 0-¹ exp(-03), where a, ß> 0 are two parameters. Show that the posterior density p(0 | X) has the same form as p(0) but with different values for a and 3. What are the new parameter values? 2. This problem copes with a new distribution: exponential distribution. Let X {x(e) € R}/₁ be an i.i.d. sample of N univariate instances drawn from an exponential density function: P(x(0) | 0) = { 0 exp(-0z(0) _r(0) 20 0 x(e) < 0, where > 0 is called the rate parameter of the distribution. (a) Find a point estimate of using maximum likelihood estimation. Show all the steps in your derivation. (b) Suppose is now treated as a random variable (instead of a fixed parameter as above) with prior density p(0) of the following form: p(0) x 0-¹ exp(-03), where a, ß> 0 are two parameters. Show that the posterior density p(0 | X) has the same form as p(0) but with different values for a and 3. What are the new parameter values? 2. This problem copes with a new distribution: exponential distribution. Let X {x(e) € R}/₁ be an i.i.d. sample of N univariate instances drawn from an exponential density function: P(x(0) | 0) = { 0 exp(-0z(0) _r(0) 20 0 x(e) < 0, where > 0 is called the rate parameter of the distribution. (a) Find a point estimate of using maximum likelihood estimation. Show all the steps in your derivation. (b) Suppose is now treated as a random variable (instead of a fixed parameter as above) with prior density p(0) of the following form: p(0) x 0-¹ exp(-03), where a, ß> 0 are two parameters. Show that the posterior density p(0 | X) has the same form as p(0) but with different values for a and 3. What are the new parameter values?

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