Suppose that your wife is pregnant and due in 100 days from

Suppose that your wife is pregnant and due in 100 days from today. Suppose that the probability density function for having a child is approximately normal with mean 100 and standard deviation 8. You have a business trip and will return in 85 days from today and have to go on another business trip in 107 days from today.

(a) What is the probability that the birth will occur before your second trip?

(b) What is the probability that the birth will occur after you return from your first business trip?

(c) What is the probability that you will be there for the birth?

(d) Given that the birth does not occur before your first business trip, what is the probability that the birth will occur before your second trip?

(e) You are able to cancel your second business trip, and your boss tells you that you can return home from your first trip if there is a 99% chance that you will make it back for the birth. When must you return home?

Solution

let X be the random variable denoting the birth of the child.

X~N(100,8²) approximately.   and Z=(X-100)/8~N(0,1)

You have a business trip and will return in 85 days from today and have to go on another business trip in 107 days from today.

a)P[the birth will occur before second trip]=P[X<107]=P[(X-100)/8<(107-100)/8]=P[Z<0.875]=0.809213 [answer]

b) P[the birth will occur after return from first trip]=P[X>85]=P[Z>(85-100)/8]=P[Z>-1.875]=0.9696036 [answer]

c) P[you will be there for the birth]=P[85<X<107]=P[X<107]-P[X<85]=P[Z<0.875]-{1-P[X>85]}

                                                                      =P[Z<0.875]-{1-P[Z>-1.875]}=0.809213-{1-0.9696036}=0.7788166 [answer]

d) Pgiven that the birth does not occur before your first business trip, but the birth will occur before your second trip]

=P[X<107|x>85]=P[85<X<107]/P[X>85]=0.7788166/0.9696036=0.803231959 [answer]

e) let you must return within k days.

as given by the question.

P[X>k]=0.99 or P[X<k]=0.01 or P[Z<(k-100)/8]=0.01=P[Z<-2.32635]

therefore (k-100)/8=-2.32635 or, k=81.3892=82 [approx]

hence he must return with in 81 days. [answer]