Zainab runs her own business where she leads guided day-long tours for small groups of tourists in her city. Each tour has a maximum of 3 guests. Let $X$ represent the number of customers Zainab gets on a randomly chosen day. Based on previous data, here is the probability distribution of $X$ along with summary statistics:
\begin{tabular}{lrrrr}
\hline$X=$ \# of customers & 0 & 1 & 2 & 3 \
$P(X)$ & 0.20 & 0.10 & 0.40 & 0.30 \
& & & Mean: & $\mu_{X}=1.8$ \
& & & Standard deviation: & $\sigma_{X} \approx 1.08$
\end{tabular}

It costs Zainab $\$ 5$ each day to pay for her own transportation to her business, and then each customer she gets pays her $\$ 60$. Let $Y$ represent Zainab's net gain on a randomly chosen day.

What are the mean and standard deviation of $Y$ ?
$$
\begin{array}{ll}
\mu_{Y}=\square \text { dollars } \
\sigma_{Y} \approx \square \text { dollars }
\end{array}
$$

Question

Zainab runs her own business where she leads guided day-long tours for small groups of tourists in her city. Each tour has a maximum of 3 guests. Let $X$ represent the number of customers Zainab gets on a randomly chosen day. Based on previous data, here is the probability distribution of $X$ along with summary statistics:
\begin{tabular}{lrrrr}
\hline$X=$ \# of customers & 0 & 1 & 2 & 3 \
$P(X)$ & 0.20 & 0.10 & 0.40 & 0.30 \
& & & Mean: & $\mu_{X}=1.8$ \
& & & Standard deviation: & $\sigma_{X} \approx 1.08$
\end{tabular}

It costs Zainab $\$ 5$ each day to pay for her own transportation to her business, and then each customer she gets pays her $\$ 60$. Let $Y$ represent Zainab's net gain on a randomly chosen day.

What are the mean and standard deviation of $Y$ ?
$$
\begin{array}{ll}
\mu_{Y}=\square \text { dollars } \
\sigma_{Y} \approx \square \text { dollars }
\end{array}
$$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Calculate the mean of $Y$ using the formula $\mu_Y = \mu_X \cdot 60 - 5$. ‖ ‖ step 2 ⋮ Substitute the given mean of $X$ ($\mu_X = 1.8$) into the formula: $\mu_Y = 1.8 \cdot 60 - 5$. ‖ ‖ step 3 ⋮ Perform the calculation: $\mu_Y = 108 - 5$. ‖ ‖ step 4 ⋮ Simplify to find the mean of $Y$: $\mu_Y = 103$. ‖ ‖ step 5 ⋮ Calculate the standard deviation of $Y$ using the formula $\sigma_Y = \sigma_X \cdot 60$. ‖ ‖ step 6 ⋮ Substitute the given standard deviation of $X$ ($\sigma_X \approx 1.08$) into the formula: $\sigma_Y = 1.08 \cdot 60$. ‖ ‖ step 7 ⋮ Perform the calculation to find the standard deviation of $Y$: $\sigma_Y \approx 64.8$. ‖ ∻[1] Answer∻ ⚹ $\mu_Y = 103$ dollars ⚹ ⚹ $\sigma_Y \approx 64.8$ dollars ⚹ ∻Key Concept∻ ⚹Linear transformation of a random variable⚹ ∻Explanation∻ ⚹The mean of a linearly transformed random variable $Y = aX + b$ is given by $\mu_Y = a\mu_X + b$, and the standard deviation is given by $\sigma_Y = |a|\sigma_X$, where $a$ and $b$ are constants. In this case, $a = 60$ and $b = -5$.⚹◊What is the expected value of the net gain variable $Y$ for Zainab's tour business?⍭ Generate me a similar question◊

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