Question

consider the probability distribution shown for the random variable x found below. complete part a through f.
x 5 6 8 14
p(x) 0.4 0.2 0.2 0.2
a. find \\(\mu = e(x)\\).
\\(\mu = 7.6\\) (round to the nearest tenth as needed )
b. find \\(\sigma^2 = e\left(x - \mu)^2\
ight\\)
\\(\sigma^2 = 11.44\\) (round to the nearest hundredth as needed )
c. find \\(\sigma\\)
\\(\sigma = \square\\) (round to four decimal places as needed )

Explanation:

Step1: Recall the formula for standard deviation

The standard deviation \(\sigma\) is the square root of the variance \(\sigma^2\), so \(\sigma = \sqrt{\sigma^2}\).

Step2: Substitute the given variance

We know that \(\sigma^2 = 11.44\), so we calculate \(\sigma=\sqrt{11.44}\).

Step3: Compute the square root

Using a calculator, \(\sqrt{11.44} \approx 3.3823\) (rounded to four decimal places).

Answer:

\(3.3823\)