Answer:
(a) The variance and standard deviation of <em>Z</em> = 35 - 10<em>X</em> are 1600 and 40 respectively.
(b) The variance and standard deviation of <em>Z</em> = 12<em>X</em> - 5 are 2304 and 48 respectively.
(c) The variance and standard deviation of <em>Z</em> = <em>X</em> + <em>Y</em> are 80 and 8.94 respectively.
(d) The variance and standard deviation of <em>Z</em> = <em>X</em> - <em>Y</em> are 80 and 8.94 respectively.
(e) The variance and standard deviation of <em>Z</em> = -2<em>X</em> + 2<em>Y</em> are 320 and 17.89 respectively.
Step-by-step explanation:
The random variable <em>X</em> has mean and standard deviation as follows:
The random variable <em>Y</em> has mean and standard deviation as follows:
It is provided that the correlation between X and Y is 0.
This implies that Cov (X, Y) = 0.
(a)
Compute the variance of <em>Z</em> = 35 - 10<em>X</em> as follows:
Then the standard deviation of <em>Z</em> = 35 - 10<em>X </em>is:
Thus, the variance and standard deviation of <em>Z</em> = 35 - 10<em>X</em> are 1600 and 40 respectively.
(b)
Compute the variance of <em>Z</em> = 12<em>X</em> - 5 as follows:
Then the standard deviation of <em>Z</em> = 12<em>X </em>- 5 is:
Thus, the variance and standard deviation of <em>Z</em> = 12<em>X</em> - 5 are 2304 and 48 respectively.
(c)
Compute the variance of <em>Z</em> = <em>X</em> + <em>Y</em> as follows:
Then the standard deviation of <em>Z</em> = <em>X</em> + <em>Y</em> is:
Thus, the variance and standard deviation of <em>Z</em> = <em>X</em> + <em>Y</em> are 80 and 8.94 respectively.
(d)
Compute the variance of <em>Z</em> = <em>X</em> - <em>Y</em> as follows:
Then the standard deviation of <em>Z</em> = <em>X</em> - <em>Y</em> is:
Thus, the variance and standard deviation of <em>Z</em> = <em>X</em> - <em>Y</em> are 80 and 8.94 respectively.
(e)
Compute the variance of <em>Z</em> = -2<em>X</em> + 2<em>Y</em> as follows:
Then the standard deviation of <em>Z</em> = -2<em>X</em> + 2<em>Y</em> is:
Thus, the variance and standard deviation of <em>Z</em> = -2<em>X</em> + 2<em>Y</em> are 320 and 17.89 respectively.