# Distribution, mean, and standard deviation of a random variable

A pair of dice is tossed. The sample space S consists of the 36 ordered pairs (a, b) where a and b can be any of the integers from 1 to 6. Let Y assign to each point the maximal of the two numbers; then Y is a random variable with range space $$R_y = 1, 2, 3, 4, 5, 6$$

Would this be correct?

1. Find the distribution of the random variable Y.

$$\begin{array}{|c|c|} \hline \text{Y}&\text{1} & \text{2}& \text{3}& \text{4}& \text{5}& \text{6} \\ \hline \text{f(Y)} & \frac{1}{36} & \frac{3}{36} & \frac{5}{36} & \frac{7}{36} & \frac{9}{36} & \frac{11}{36} \\ \hline \end{array}$$

1. Find the expectation of the random variable Y.

$$1(\frac{1}{36}) + 2(\frac{3}{36})+3(\frac{5}{36})+4(\frac{7}{36})+5(\frac{9}{36})+6(\frac{11}{36}) = \frac{161}{36}$$

1. Find the standard deviation of the random variable Y.

$$\sigma = \sqrt \frac{161}{36}$$

• The standard deviation is not defined as the mean's root. Mar 15, 2017 at 12:42

The variance of a random variable X is defined as E[$X^2$] - $(E[X])^2$.
You have calculated E[X], now you need to calculate $E[X^2]$.