Chapter 5, Problem 59SE

The U.S. Coast Guard (USCG) provides a wide variety of information on boating accidents including the wind condition at the time of the accident. The following table shows the results obtained for 4401 accidents (USCG website, November 8, 2012).

Wind Condition	Percentage of Accidents
None	9.6
Light	57.0
Moderate	23.8
Strong	7.7
Storm	1.9

Let x be a random variable reflecting the known wind condition at the time of each accident. Set x = 0 for none, x = 1 for light, x = 2 for moderate, x = 3 for strong, and x = 4 for storm.

1. a. Develop a probability distribution for x.
2. b. Compute the expected value of x.
3. c. Compute the variance and standard deviation for x.

Comment on what your results imply about the wind conditions during boating accidents.

To determine

Construct a probability distribution for the random variable x.

Answer to Problem 59SE

The probability distribution for the random variable x is given by,

x	$f\left(x\right)$
0	0.0960
1	0.05700
2	0.2380
3	0.0770
4	0.0190

Explanation of Solution

Calculation:

The data represents the results obtained for 4,401 boating accidents including the wind condition at the time of the accident. The random variable x represents the known wind condition at the time of each accident. The random variable x takes the value 0 for none,

takes the value 1 for light, takes the value 2 for moderate, takes the value 3 for strong, takes the value 4 for storm.

Here, the total number of responses is 4,401. The corresponding probabilities are obtained by converting the percentages in to probabilities. That is, by dividing each value with 100.

The probability distribution for the random variable x can be obtained as follows:

x	f	$\frac{f}{N}$	$f\left(x\right)$
0	9.6	$\frac{9.6}{100}$	0.0960
1	57.0	$\frac{57.0}{100}$	0.5700
2	23.8	$\frac{23.8}{100}$	0.2380
3	7.7	$\frac{7.7}{100}$	0.0770
4	1.9	$\frac{1.9}{100}$	0.0190
Total	100		1

To determine

Find the expected value for the random variable x.

Answer to Problem 59SE

The expected value for the random variable x is 1.35.

Explanation of Solution

Calculation:

The formula for the expected value of a discrete random variable is,

$\begin{array}{c}E\left(x\right)=\mu \\ ={\displaystyle \sum x\cdot f\left(x\right)}\end{array}$.

The expected value for the random variable x is obtained using the following table:

x	f(x)	$x\cdot f\left(x\right)$
0	0.096	0
1	0.57	0.57
2	0.238	0.476
3	0.077	0.231
4	0.019	0.076
Total	1	1.353

Thus, the expected value for the random variable x is 1.353.

To determine

Find the variance and standard deviation of the random variable x.

Answer to Problem 59SE

The variance of the random variable x is 0.6884.

The standard deviation of the random variable x is 0.8297.

Explanation of Solution

Calculation:

The formula for the variance of the discrete random variable is,

${\sigma }^2={\displaystyle \sum \left[{\left(x-\mu \right)}^2.f\left(x\right)\right]}$.

The variance of the random variable x is obtained using the following table:

x	f(x)	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$	${\left(x-\mu \right)}^2.f\left(x\right)$
0	0.096	–1.353	1.8306	0.1757
1	0.57	–0.353	0.1246	0.0710
2	0.238	0.647	0.4186	0.0996
3	0.077	1.647	2.7126	0.2089
4	0.019	2.647	7.0066	0.1331
Total	1	3.235	12.0930	0.6884

Therefore,

${\sigma }^2=0.6884$

Thus, the variance of the random variable x is 0.6884.

The formula for the standard deviation of the discrete random variable is,

$\sigma =\sqrt{{\displaystyle \sum \left[{\left(x-\mu \right)}^2.f\left(x\right)\right]}}$.

Thus, the standard deviation is,

$\begin{array}{c}\sigma =\sqrt{0.6884}\\ =0.8297\end{array}$

Hence, the standard deviation of the random variable x is 0.8297.

To determine

Explain the result that implies about the wind conditions during the boating accidents.

Explanation of Solution

The expected value is 1.353 and it represents the average wind conditions during accident. This value is slightly less than light wind conditions.