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# Combining normal random variables

When we combine variables that each follow a normal distribution, the resulting distribution is also normally distributed. This lets us answer interesting questions about the resulting distribution.

## Example 1: Total amount of candy

Each bag of candy is filled at a factory by $4$ machines. The first machine fills the bag with blue candies, the second with green candies, the third with red candies, and the fourth with yellow candies. The amount of candy each machine dispenses is normally distributed with a mean of $50\phantom{\rule{0.167em}{0ex}}\text{g}$ and a standard deviation of $5\phantom{\rule{0.167em}{0ex}}\text{g}$. Also, assume that the amount dispensed by any given machine is independent from the other machines.
Let $T$ be the total weight of candy in a randomly selected bag.
Find the probability that a randomly selected bag contains less than $178\phantom{\rule{0.167em}{0ex}}\text{g}$ of candy.
Let's solve this problem by breaking it into smaller pieces.
Problem A (Example 1)
Find the mean of $T$.
${\mu }_{T}=$
grams

Problem B (Example 1)
Find the standard deviation of $T$.
${\sigma }_{T}=$
grams

Problem C (Example 1)
What shape does the distribution of $T$ have?

Problem D (Example 1)
Find the probability that a randomly selected bag contains less than $178\phantom{\rule{0.167em}{0ex}}\text{g}$ of candy.
Round to four decimal places.
$P\left(T<178\phantom{\rule{0.167em}{0ex}}\text{g}\right)\approx$

## Example 2: Difference in bowling scores

Adam and Mike go bowling every week. Adam's scores are normally distributed with a mean of $175$ pins and a standard deviation of $30$ pins. Mike's scores are normally distributed with a mean of $150$ pins and a standard deviation of $40$ pins. Assume that their scores in any given game are independent.
Let $A$ be Adam's score in a random game, $M$ be Mike's score in a random game, and $D$ be the difference between Adam's and Mike's scores where $D=A-M$.
Find the probability that Mike scores higher than Adam in a randomly selected game.
Let's solve this problem by breaking it into smaller pieces.
Problem A (Example 2)
Find the mean of $D$.
${\mu }_{D}=$
pins

Problem B (Example 2)
Find the standard deviation of $D$.
${\sigma }_{D}=$
pins

Problem C (Example 2)
What shape does the distribution of $D$ have?

Problem D (Example 2)
Find the probability that Mike scores higher than Adam in a randomly selected game.
Round to four decimal places.
$P\left(\text{Mike scores higher}\right)\approx$
Hint: Find $P\left(D<0\right)$.

## Want to join the conversation?

• In Example 2: The hint says P(D < 0), why the probability of the difference between the two data has to be less than 0?
• We have D = A - M. If D < 0, then it can happen only when M > A, which means Mike scores higher than Adam.
P(D < 0) means probability of an event where Mike scores higher than Adam.
Hope that helps.
• In example 2 the number of pins is discrete, how could you represent that using a density curve ?
• In the Practice quiz they keep having an absolute value probability question. How does one go about solving that? For instance one example is Sam's mean of washing cars is 20 minutes with a standard deviation of 6.4 minutes. Taylor's mean of washing the interior of cars is 18 minutes with a mean of 4.8 minutes.
Then it says find the probability that a randomly selected time of Sam and Taylor falls within 10 minutes of each other and gives the equation find P(D less than |10|). So I do D=(S-T) and I get mean of D is 2 minutes and the standard deviation is 8 minutes. So far so good, but after that I always go wrong somehow. When I click on the explanation it says to do two z scores one of -10 and one of 10 and then calculate between them, but why would I do -10 and 10, it says within ten minutes of each other, wouldn't that mean you would do ten above and ten below the mean of D?
• The mean and standard deviation explain the shape of the curve and can tell which percentages are above and below certain points. However, the question asks whether they finish within 10 minutes of each other. Since Taylor is 2 minutes quicker than Sam, the area under the curve is shifted. The center where they both have the same time is 0, reflecting where both Sam and Taylor have the same finish time (S-T). Calculate 10 minutes below and 10 minutes above 0, the place where they are equal, to find the percentages where they are finishing within 10 minutes of each other.
(1 vote)
• Hiya Sal and everyone at Khan, thank you for all your hard work. It would be really nice if we could get a worked example of a probability of an absolute value as that is something that comes up in the practice questions but wasn't covered in the videos leading up to it. Like P(X |5|) or something like that.
• P(X = |5|) = P(X = 5)

I think you meant something different, since you can always just replace |5| with 5.

If you meant to ask something like
P(|X| > 5)
this would be (I think)
P(|X| > 5) = P(X < -5) + P(X > 5)