## Question 623:

1

The mean of a random variable is the expected value over the long term. To find it you simply sum the values of x times their probability of occurrence. Since we're given the actual number of hurricanes in each category, we also need to calculate the probability of each type. This will make up our probability distribution. We do this by simply dividing the raw number by the total.

 Hurricane Type X P 1 57 0.360759 2 37 0.234177 3 47 0.297468 4 15 0.094937 5 2 0.012658

Now we multiply the probability by the category, which gets us our expected value and mean.

1(.36) + 2(.23) + 3(.297) + 4(.094) + 5(.012) = 2.164 is our mean

To find the standard deviation, you follow the same procedure you would if the variable was continuous. Subtract each value from the mean (the value you just calculated), square the difference, then multiply the probability times that results. Finally take the square root of the result.

(1-2.164)2 * .36  +  (2-2.164)2 * .23  +  (3-2.164)2 * .297  +  (4-2.164)2 * .094  + (5-2.164)2 * .012 =   2.687

So 2.687 is the variance of the distribution, the square root of 2.687 = 1.639 and is the standard deviation

Finally, we're asked what the probability of at least a category 4 hurricane. To find this, we simply add up the probability of 4 and 5, which gets us .094+.0126 = .1075. So the probability of a category 4 or more hurricane is .1075, which is about 10.75%.