Suppose you are playing a game. The random variable, Y, represents your net profit in dollars. The probability distribution of Y is as follows. (Round your answers to 4 decimal places, if needed.)

Question

Suppose you are playing a game. The random variable, Y, represents your net profit in dollars. The probability distribution of Y is as follows. (Round your answers to 4 decimal places, if needed.)

y	-3	-2	-1	3	5
P(Y = y)	0.1	0.23	?	0.12	0.15

(a) What is the mean for net profit?

(b) What is the median for net profit?

(c) What is the standard deviation for net profit?

Answer 1

The first thing you need to do is calculate the missing probability: P(y=-1). Since the sum of all the probabilities must =1, then add the probabilities given, then subtract that number from 1.

We're told that Y is the variable for net profit, and we have the complete distribution, so calculate the mean by multiplying the value of Y by its probability: P(Y=y), then add them all together, and that's the mean!

To find the median, takes a little more work. The median of a discrete random variable is the "middle" value. It is the value of X for which P(X < x) is greater than or equal to 0.5 and P(X > x) is greater than or equal to 0.5. You'll just need to play a bit to find it.

As for the standard deviation: First, find the expected value; then, find the variance; then, find the standard deviation. The expected value is also known as the mean, which you've already calculated. Now that we know the expected value, we find the variance.

σ² = Σ { [ x_i - E(x) ]² * P(x_i) }

And finally, the standard deviation is equal to the square root of the variance;