A neighborhood contains 21 single family homes of which 4 ho

A neighborhood contains 21 single family homes of which 4 homes have no children, 3 homes
have 1 child, 8 homes have 2 children, 4 homes have 3 children, 1 home has 4 children, and 1
home has six children. One of the families will be randomly selected to win a hula hoop for each
child living in the house (one hula hoop per child). Let X=the number of hula hoops that will be
won.
a) Make a table that gives the probability mass function for X.
b) Give the cumulative distribution function for X.
c) What is the expected number of hula hoops that will be won?
d) What is the variance for the number of hula hoops that will be won?
e) If hula hoops cost $6 each, what is the expected total cost of the hula hoops that will be
won? (Note that you should be able to figure this out using your answer to part c.)
f) What is the variance of the total cost of the hula hoops that will be won? (Note that you
should be able to figure this out using your answer to part d.)
g) What is the probability that less than $18 will be spent on the hula hoops?

Please do a step by step walk through

Solution

a)

PMF

b)

cumulative distribution

c)

So, expected number of hula hoops = 1.9523

d)

Variance for number of hula hoops = 0.3408

e)

Expected cost = 6*1.9523 = 11.71

f)

Variance = 36*0.3408 = 12.2688

g)

the probability that less than $18 will be spent on the hula hoops

= Probability that less than 3 hula hoops will be won

= P( x<3) =P(x <=2)

= 0.714