Question

A normally distributed random variable with mean 4.5 and standard deviation 7.6 is sampled to get two independent values, X1 and X2. The mean is estimated using the formula (3X1 4X2)/8. Determine the bias and the mean squared error for this estimator of the mean. Bias: Mean Square Error:

Answer 1

Answer:

Bias for the estimator = -0.56

Mean Square Error for the estimator = 6.6311

Step-by-step explanation:

Given - A normally distributed random variable with mean 4.5 and standard deviation 7.6 is sampled to get two independent values, X1 and X2. The mean is estimated using the formula (3X1 + 4X2)/8.

To find - Determine the bias and the mean squared error for this estimator of the mean.

Proof -

Let us denote

X be a random variable such that X ~ N(mean = 4.5, SD = 7.6)

Now,

An estimate of mean, μ is suggested as

$\mu = \frac{3X_{1} + 4X_{2} }{8}$

Now

Bias for the estimator = E(μ bar) - μ

                                    = $E( \frac{3X_{1} + 4X_{2} }{8}) - 4.5$

                                    = $\frac{3E(X_{1}) + 4E(X_{2})}{8} - 4.5$

                                    = $\frac{3(4.5) + 4(4.5)}{8} - 4.5$

                                    = $\frac{13.5 + 18}{8} - 4.5$

                                    = $\frac{31.5}{8} - 4.5$

                                    = 3.9375 - 4.5

                                    = - 0.5625 ≈ -0.56

∴ we get

Bias for the estimator = -0.56

Now,

Mean Square Error for the estimator = E[(μ bar - μ)²]

                                                             = Var(μ bar) + [Bias(μ bar, μ)]²

                                                             = $Var( \frac{3X_{1} + 4X_{2} }{8}) + 0.3136$

                                                             = $\frac{1}{64} Var( {3X_{1} + 4X_{2} }) + 0.3136$

                                                             = $\frac{1}{64} ( [{3Var(X_{1}) + 4Var(X_{2})] }) + 0.3136$

                                                             = $\frac{1}{64} [{3(57.76) + 4(57.76)}] } + 0.3136$

                                                             = $\frac{1}{64} [7(57.76)}] } + 0.3136$

                                                             = $\frac{1}{64} [404.32] } + 0.3136$

                                                             = $6.3175 + 0.3136$

                                                              = 6.6311

∴ we get

Mean Square Error for the estimator = 6.6311