Question

Math

Posted 4 months ago

```
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

To find the standard deviation $\sigma_X$, we first need to calculate the variance $\sigma_X^2$. The variance is given by the formula $\sigma_X^2 = E[X^2] - (\mu_X)^2$, where $E[X^2]$ is the expected value of the square of X

step 2

Calculate $E[X^2]$ using the probability distribution. $E[X^2] = (100^2 \times 0.10) + (10^2 \times 0.90)$

step 3

Simplify the calculation of $E[X^2]$. $E[X^2] = (10000 \times 0.10) + (100 \times 0.90) = 1000 + 90 = 1090$

step 4

Use the given mean $\mu_X = 19$ to calculate the variance. $\sigma_X^2 = E[X^2] - (\mu_X)^2 = 1090 - (19)^2$

step 5

Simplify the calculation of the variance. $\sigma_X^2 = 1090 - 361 = 729$

step 6

Calculate the standard deviation $\sigma_X$ by taking the square root of the variance. $\sigma_X = \sqrt{729}$

step 7

Simplify the calculation of the standard deviation. $\sigma_X = 27$

Answer

$\sigma_{X}=27$ points

Key Concept

Standard Deviation and Variance of a Probability Distribution

Explanation

The standard deviation is the square root of the variance, which measures the spread of a probability distribution around the mean. It is calculated using the expected value of the square of the random variable and the square of the mean.

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