Suppose we have a six-sided die, but it is weighted so that the probabilities of any face showing are not all the same. Let X be a random variable for the number showing on the die. The probabilities are as follows:

X P(x)

1 0.3

2 0.05

3 0.1

4 0.1

5 0.05

6 0.4

Compute the expected value (mean) and the variance of the random variable, X.

If the die was a fair die (1/6 probability for each face), the expected value would be 3.5 and the variance would be approximately 2.92. Your answer to the question above should be different. Is the variance you found higher or lower? Does this make intuitive sense? Explain.