In Exercises 7–14, determine whether a probability

distribution is given. If a probability distribution is given, find its mean and standarddeviation. If a probability distribution is not given, identify the requirements that are not satisfied.

When conducting research on color blindness in males, a researcher forms random groups with fivemales in each group. The random variable xis the number of males in the group who have a form of color blindness (based on data from the National Institutes of Health).

x	P(x)
0	0.659
1	0.287
2	0.05
3	0.004
4	0.001
5	0+

The probability distribution for color blindness in males is provided.

The variable x is the number of males in the group who have a form of color blindness.

The requirements are as follows.

1)The variable x isa numerical random variable.

2)The sum of the probabilities is computed as

$$\begin{gathered} \sum P\left(x\right)=0.659+0.287+0.05+0.004+0.001+0.000 \\ =1.001 \end{gathered}$$

Therefore,the sum of the probabilities is approximately equal to 1 with a round-off error of 0.001.

3) Each value of P(x) is between 0 and 1.

Thus, the probability distribution is valid.

The mean for the random variable is computed as

$$\begin{gathered} \mu =\sum \left[x\times P\left(x\right)\right] \\ =\left(0\times 0.659\right)+\left(1\times 0.287\right)+\left(3\times 0.05\right)+...+\left(5\times 0.000\right) \\ =0.403 \end{gathered}$$

Thus, the mean value of the random variable x is 0.4.

The standard deviation of the random variable x is computed as

$\sigma =\sqrt{\sum \left[x^2\times P\left(x\right)\right]-{\mu }^2}$

The calculations are as follows.

$$\begin{gathered} \sum \left[x^2·P\left(x\right)\right]=\left(0^2\times 0.659\right)+\left(1^2\times 0.287\right)+\left(2^2\times 0.05\right)+...+\left(5^2\times 0.000\right) \\ =0.539 \end{gathered}$$

The standard deviation is given as

$$\begin{gathered} \sigma =\sqrt{\sum \left[x^2·P\left(x\right)\right]-{\mu }^2} \\ =\sqrt{0.539-{0.403}^2} \\ =0.6136 \\ \approx 0.6 \end{gathered}$$

Thus, the standard deviation of x is 0.6.