Question

The number of breakdowns Y per day for a certain machine is a Poisson random variable with mean A. The daily cost of repairing these breakdowns is given by C 3Y2. If Y, Y2, Y denote the observed number of breakdowns for n independently selected days, find an MVUE for E(C).

Answer 1

Answer:

Let Y be the random variable that represents the number of breakdowns per day for a certain machine Here, the random variable Y follows Poisson distribution with mean 2 The daily cost of repairing these breakdowns is given by c-3Y Consider Y,,Y,. ...Y, denote the observed number of breakdowns for n independently selected days. The objective of the problem is to find an MVUE for E(C)

Use Rao-Blackwell Theorem to solve this problem. First need to find an unbiased estimator U for E(C) and a sufficient statistic T Then the estimator C-E(U | T) is the MVUE for E(C) Consider 3yyi Therefore, In E(U) - E(C) Hence, U2Σ.2 is an unbiased estimator for E(C) In

Since y is a Poisson random variable with mean 2 The probability density function is as follows e 2 r! The likelihood function is obtained as follows: し(x, ). I2)=f(x,...-X 12) 2 Пр! 2 5 Therefore, T Y is a sufficient statistics.

Then the estimator C-E(UI) is the MUVE for E(C) /n Since E(Y|Y)-Y Therefore, the MUVE for E(C) is 3P + | |--

Thank you!