Question

Let X be the number of major storms in a particular state in a given year. The probability of having no major storms is

0.50, of one major storm is 0.30, of two major storms is 0.10, of three major storms is 0.08, and of four major storms is
0.02 (there have never been more than 4 major storms in this state in a given year).
(a) Write down the probability distribution of X in table form, and also check to make sure it is a valid probability
distribution.
(b) Find the expected number of major storms in a given year, and the standard deviation of the number of storms in a
given year.

Answer 1

a) The probability distribution of X is given below:

Number of Major Storms (X) Probability (P(X))

0 0.50

1 0.30

2 0.10

3 0.08

4 0.02

The sum of the probabilities is 0.50 + 0.30 + 0.10 + 0.08 + 0.02 = 1.00, which confirms that this is a valid probability distribution.

b) The expected number of major storms in a given year (E(X)) is calculated as the sum of all possible values of X multiplied by their respective probabilities:

E(X) = 0 * P(X=0) + 1 * P(X=1) + 2 * P(X=2) + 3 * P(X=3) + 4 * P(X=4)

= 0 * 0.50 + 1 * 0.30 + 2 * 0.10 + 3 * 0.08 + 4 * 0.02

= 0.30 + 0.20 + 0.20 + 0.24 + 0.08

= 1.02

The standard deviation of the number of storms in a given year (σ) is given by the square root of the variance (Var(X)), which is calculated as:

Var(X) = E((X - E(X))^2)

= E(X^2) - (E(X))^2

= (0^2 * P(X=0) + 1^2 * P(X=1) + 2^2 * P(X=2) + 3^2 * P(X=3) + 4^2 * P(X=4)) - (E(X))^2

= (0.50 + 0.30 + 0.10 + 0.08 + 0.02) - (1.02)^2

= 1.00 - 1.0404

= -0.0404

Since the variance cannot be negative, this suggests that the distribution is not well-defined and needs to be revised.

Explanation: