Question

Math

Posted 4 months ago

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Gloria enjoys playing Skee ball - an arcade game where players toss a ball so it rolls up a ramp and falls into one of several slots. Each slot is worth a different number of points based on how difficult it is to land the ball in that slot. Gloria only aims at the most difficult slot, which is worth 100 points, but she only has a $10 \%$ chance of landing the ball in that slot. If she misses, her ball will certainly land in the 10 point slot.
The table below displays the probability distribution of $X$, the number of points Gloria scores on a random shot.
\begin{tabular}{lll}
\hline$X=$ points & 100 & 10 \\
$P(X)$ & $10 \%$ & $90 \%$
\end{tabular}
Given that $\mu_{X}=19$ points, calculate $\sigma_{X}$.
$\sigma_{X}=$ $\square$ points
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Identify the possible outcomes and their probabilities

step 2

Use the formula for the variance of a discrete random variable, $\sigma_X^2 = \sum [P(X) \cdot (X - \mu_X)^2]$

step 3

Calculate the squared deviations from the mean for each outcome: $(100 - 19)^2$ and $(10 - 19)^2$

step 4

Multiply each squared deviation by its corresponding probability: $0.1 \cdot (100 - 19)^2$ and $0.9 \cdot (10 - 19)^2$

step 5

Sum the results to find the variance: $\sigma_X^2 = 0.1 \cdot (100 - 19)^2 + 0.9 \cdot (10 - 19)^2$

step 6

Calculate the standard deviation by taking the square root of the variance: $\sigma_X = \sqrt{\sigma_X^2}$

Answer

$\sigma_X = \sqrt{0.1 \cdot (100 - 19)^2 + 0.9 \cdot (10 - 19)^2}$ points

Key Concept

Standard Deviation of a Discrete Random Variable

Explanation

The standard deviation is the square root of the variance, which is the sum of the squared differences between each outcome and the mean, weighted by their probabilities.

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