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1. A couple decides to have children until they have a boy. Let's assume that they
are able to continue to do so and each time they conceive, they only have a single
child (no twins or triplets, etc.).
a. What is the probability that the first time that they have a boy is on
their 4th child (that is, that they have 3 girls before they have a son)?
b. If their first 3 children are girls, what is the probability that their next
child is boy?
c.
On average, how many children should they expect to have to have
before having a boy? (Hint: use math expectation)
2. We have a die that is numbered 3 with two 1's, two 2's, and two 3's. Let's
assume it is fair (each side is equally probable) and that the die is rolled 3 times.
And, let's assume that the 3 rolls are independent. Let X, Y, and Z be the
outcomes of the first, second and third rolls, respectively.
a.
What is the probability distribution of X+Y+Z? That is, create a table that
contains each unique possible value of X+Y (each value only listed once) and
each possibility's corresponding probability.
b. What is the probability that X+Y+Z is greater than or equal to 7?
3. We have a fair six-sided die numbered 2, 4, 6, 8 10, and 12.
a. Find the math expectation of a single roll.
b. Find the math expectation of the numerical sum of 5 rolls.
c. Find the math expectation of the numerical product (i.e.,
multiplication) of 4 rolls.
d. Find the cumulative distribution function of the outcome of a single
die roll and draw its graph.
4. X1. X2, ... X144 are independent and identically distributed random variables
such that E(x) = 12 and var(x) = 36. What is the standard deviation of their
average? In other words, what is the standard deviation of X = X1+X2+ +X1442
144
5. If the cumulative distribution function of X is given by the function below, then
find P (X <1.5).
F(x) 0, if S o
F(x) = =x if o x<2
F(x)=1,if x>2
6. A local pizza place has a profit of $1, $3, and $2 for each sale of their small,
medium and large pizzas, respectively. In general, their small (personal size)
and large (family size) are their most popular pizza sizes (with about 40% of
their sales contributing to each of these sizes).
Size
Small
Medium
Large
x
$1
$3
$2
P
0.40
0.20
0.40
a. Find the math expectation of a single pizza sale.
b.
In a typical day, there are about 200 pizza sales. Find the math
expectation of the profit for a single day (200 sales).
c. Find the variance of the profit for a single sale.
d. Find the standard deviation for a single sale.
c. What might the pizza place do to increase their profit?

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