91Ó°ÊÓ

The standard deviation is a statistical measure of how much a set of values varies or disperses. A low standard deviation implies that the values are close to the set's mean (also known as the anticipated value), whereas a large standard deviation shows that the values are spread out over a wider range.

The given random variable can be written as

\(Y = \frac{1}{2}{X_1} + \frac{1}{2}{X_2} - \frac{1}{3}{X_3} - \frac{1}{3}{X_4} - \frac{1}{3}{X_5}\)

It is normally distributed with mean value of

\(\begin{aligned}E(Y) &= E\left( {\frac{1}{2}{X_1} + \frac{1}{2}{X_2} - \frac{1}{3}{X_3} - \frac{1}{3}{X_4} - \frac{1}{3}{X_5}} \right)\\ &= \frac{1}{2}E\left( {{X_1}} \right) + \frac{1}{2}E\left( {{X_2}} \right) - \frac{1}{3}E\left( {{X_3}} \right) - \frac{1}{3}E\left( {{X_4}} \right) - \frac{1}{3}E\left( {{X_5}} \right)\\ &= - 1\end{aligned}\)

(1): the expected values are given in the mentioned example.

The variance of random variable Y is

\(\begin{aligned}V(Y) &= V\left( {\frac{1}{2}{X_1} + \frac{1}{2}{X_2} - \frac{1}{3}{X_3} - \frac{1}{3}{X_4} - \frac{1}{3}{X_5}} \right)\\ &= \frac{1}{4}V\left( {{X_1}} \right) + \frac{1}{4}V\left( {{X_2}} \right) + \frac{1}{9}V\left( {{X_3}} \right) + \frac{1}{9}V\left( {{X_4}} \right) + \frac{1}{9}V\left( {{X_5}} \right)\\ &= 3.167\end{aligned}\)

(2): the variances are given in the mentioned example.

The standard deviation of random variable Y is

\({\sigma _Y} = \sqrt {V(X)} = \sqrt {3.167} = 1.7795\)

The probability of event \(\{ Y \ge 0\} \)is

\(\begin{aligned}P(Y \ge 0) &= P\left( {\frac{{Y - E(Y)}}{{{\sigma _Y}}} \ge \frac{{0 - ( - 1)}}{{1.7795}}} \right)\\ &= P(Z \ge 0.56)\\ &= 1 - P(Z < 0.56)\\ &= 0..2877\end{aligned}\)

(3): from the normal probability table in the appendix. The probability can also be computed with software.

Similarly, the probability of the event \(\{ - 1 \le Y \le 1\} \)is

\(\begin{aligned}P( - 1 \le Y \le 1) &= P\left( {\frac{{ - 1 - ( - 1)}}{{1.7795}} \le Z \le \frac{{1 - ( - 1)}}{{1.7795}}} \right)\\ &= P(0 \le Z \le 1.12)\\ &= 0.3686,\end{aligned}\)

(3): from the normal probability table in the appendix. The probability can also be computed with a software.