Question

Consider a sample of $N_x$ independent random numbers $\{x_n\}$; $n = 1, N_x$ drawn from a distribution with the mean $\theta_x$ and variance $\sigma_x^2$ and another independent sample of $N_y$ random numbers $\{y_n\}$; $n = 1, N_y$ drawn from a distribution with the mean $\theta_y$ and variance $\sigma_y^2$. Form a new sample $\{z_n\}$; $n = 1, N_z$ of the size $N_z = N_x + N_y$ in this way: the first $N_x$ elements of $\{z_n\}$ are the same as the elements of $\{x_n\}$ multiplied by the ratio $N_z/N_x$ and the last $N_y$ elements of $\{z_n\}$ are the same as the elements of $\{y_n\}$ multiplied by the ratio $-N_z/N_y$ (with a minus sign), viz. $\{z_n\} = \{\frac{N_z}{N_x}x_1, \frac{N_z}{N_x}x_2, ..., \frac{N_z}{N_x}x_{N_x}, -\frac{N_z}{N_y}y_1, -\frac{N_z}{N_y}y_2, ..., -\frac{N_z}{N_y}y_{N_y}\}$. Find the expectation and the variance of the distribution the average $a_z = \frac{1}{N_z}\sum_{n=1}^{N_z}z_n$ of the sample $\{z_n\}$ is associated with. In other words, find the expectation $E\{a_z\}$ and the variance $Var\{a_z\}$. The answers should be expressed via $\theta_x, \theta_y, \sigma_x^2, \sigma_y^2, N_x$ and $N_y$ (all known numbers). Hint: first express $a_z$ via $a_x$ and $a_y$ then make use of the results derived in class for the mean and variance of (a distribution of) the average of a data sample, as well as that of the sum/difference of random numbers.

          Consider a sample of $N_x$ independent random numbers $\{x_n\}$; $n = 1, N_x$ drawn from a
distribution with the mean $\theta_x$ and variance $\sigma_x^2$ and another independent sample of $N_y$ random
numbers $\{y_n\}$; $n = 1, N_y$ drawn from a distribution with the mean $\theta_y$ and variance $\sigma_y^2$. Form a
new sample $\{z_n\}$; $n = 1, N_z$ of the size $N_z = N_x + N_y$ in this way: the first $N_x$ elements of $\{z_n\}$
are the same as the elements of $\{x_n\}$ multiplied by the ratio $N_z/N_x$ and the last $N_y$ elements of
$\{z_n\}$ are the same as the elements of $\{y_n\}$ multiplied by the ratio $-N_z/N_y$ (with a minus sign),
viz. $\{z_n\} = \{\frac{N_z}{N_x}x_1, \frac{N_z}{N_x}x_2, ..., \frac{N_z}{N_x}x_{N_x}, -\frac{N_z}{N_y}y_1, -\frac{N_z}{N_y}y_2, ..., -\frac{N_z}{N_y}y_{N_y}\}$. Find the expectation and the
variance of the distribution the average $a_z = \frac{1}{N_z}\sum_{n=1}^{N_z}z_n$ of the sample $\{z_n\}$ is associated with.
In other words, find the expectation $E\{a_z\}$ and the variance $Var\{a_z\}$. The answers should be
expressed via $\theta_x, \theta_y, \sigma_x^2, \sigma_y^2, N_x$ and $N_y$ (all known numbers). Hint: first express $a_z$ via $a_x$ and $a_y$
then make use of the results derived in class for the mean and variance of (a distribution of) the
average of a data sample, as well as that of the sum/difference of random numbers.

consider a sample of nx independent random numbers xn n 1 nx drawn from a distribution with the mean thetax and variance sigmax2 and another independent sample of ny random numbers yn 65217

Added by Derrick H.

Question

Please give Ace some feedback