a factory manager collected data on the number of equipment breakdowns per day. from those data, she derived the probability distribution shown to the right, where w denotes the number of breakdowns on a given day. answer parts a through c.
a. determine $ \mu _ { w } $ and $ \sigma _ { w } $.
$ \mu _ { w } = 0.35 $ (type an integer or a decimal.)
$ \sigma _ { w } = 0.572 $ (round to three decimal places as needed.)
b. on average, how many breakdowns occur per day?
$ \square $ (type an integer or a decimal.)

Question

Accepted Answer

# Explanation:
## Step1: Calculate the mean ($\mu_{W}$)
The formula for the mean of a discrete random variable is $\mu_{W}=\sum_{w}w\times P(W = w)$.
$$
\begin{align*}
\mu_{W}&=(0\times0.70)+(1\times0.25)+(2\times0.05)\
&=0 + 0.25+0.1\
&=0.35
\end{align*}
$$
## Step2: Calculate the variance ($\sigma_{W}^{2}$)
The formula for the variance of a discrete random variable is $\sigma_{W}^{2}=\sum_{w}(w-\mu_{W})^{2}\times P(W = w)$.
$$
\begin{align*}
\sigma_{W}^{2}&=(0 - 0.35)^{2}\times0.70+(1 - 0.35)^{2}\times0.25+(2 - 0.35)^{2}\times0.05\
&=(0.1225\times0.70)+(0.4225\times0.25)+(2.7225\times0.05)\
&=0.08575+0.105625 + 0.136125\
&=0.3275
\end{align*}
$$
## Step3: Calculate the standard deviation ($\sigma_{W}$)
The standard deviation is the square - root of the variance, $\sigma_{W}=\sqrt{\sigma_{W}^{2}}$.
$$
\sigma_{W}=\sqrt{0.3275}\approx0.572
$$

# Answer:
a. $\mu_{W}=0.35$
b. $\sigma_{W}\approx0.572$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the mean ($\mu_{W}$)

Step2: Calculate the variance ($\sigma_{W}^{2}$)

Step3: Calculate the standard deviation ($\sigma_{W}$)

Answer: