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 $Y$ ?
$\mu_{Y}=$ $\square$ dollars
$\sigma_{Y} \approx$ $\square$ dollars

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 $Y$ ?
$\mu_{Y}=$ $\square$ dollars
$\sigma_{Y} \approx$ $\square$ dollars

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the mean of $Y$, we use the formula $\mu_Y = \mu_X \cdot 60 - 5$. ‖ ‖ step 2 ⋮ Substituting the given mean of $X$, $\mu_X = 1.8$, we get $\mu_Y = 1.8 \cdot 60 - 5$. ‖ ‖ step 3 ⋮ Calculating the mean of $Y$, $\mu_Y = 108 - 5 = 103$. ‖ ‖ step 4 ⋮ To find the standard deviation of $Y$, we use the formula $\sigma_Y = \sigma_X \cdot 60$. ‖ ‖ step 5 ⋮ Substituting the given standard deviation of $X$, $\sigma_X = 1.08$, we get $\sigma_Y = 1.08 \cdot 60$. ‖ ‖ step 6 ⋮ Calculating the standard deviation of $Y$, $\sigma_Y = 64.8$. ‖ ∻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 $a$ is the multiplier and $b$ is the constant added.⚹◊What is the formula for calculating the standard deviation of Y based on the given information about X and Zainab's cost and revenue structure⍭ Generate me a similar question◊

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