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 μw and σw.
μw = .35 (type an integer or a decimal.)
σw = □ (round to three decimal places as needed.)

Question

Accepted Answer

# Explanation:
## Step1: Calculate the mean $\mu_W$
The formula for the mean of a discrete random variable is $\mu_W=\sum_{w}w\times P(W = w)$.
Here, when $w = 0$, $P(W=0)=0.70$; when $w = 1$, $P(W = 1)=0.25$; when $w=2$, $P(W = 2)=0.05$.
$$
\begin{align*}
\mu_W&=(0\times0.70)+(1\times0.25)+(2\times0.05)\
&=0 + 0.25+0.10\
&=0.35
\end{align*}
$$
## Step2: Calculate the variance $\sigma_{W}^{2}$
The formula for the variance of a discrete random variable is $\sigma_{W}^{2}=\sum_{w}(w-\mu_W)^{2}\times P(W = w)$.
$$
\begin{align*}
\sigma_{W}^{2}&=(0 - 0.35)^{2}\times0.70+(1 - 0.35)^{2}\times0.25+(2 - 0.35)^{2}\times0.05\
&=( - 0.35)^{2}\times0.70+(0.65)^{2}\times0.25+(1.65)^{2}\times0.05\
&=(0.1225)\times0.70+(0.4225)\times0.25+(2.7225)\times0.05\
&=0.08575+0.105625 + 0.136125\
&=0.3275
\end{align*}
$$
## 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:
$\mu_W = 0.35$, $\sigma_W\approx0.572$

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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: