Question

In this problem, we assume that the odds of giving birth to a girl or to a boy are 1 2 each. We consider a country in which, because of tradition and particular socio-economic circumstances, parents give birth to children until they give birth to their first son, at which point they stop having children. The point of the problem is to evaluate the proportion of children who are boys in a given generation. We will do this in several steps. A (3 Points) Let X be the random variable corresponding to the number of children of a given couple chosen at random in the population. What is the probability mass function of X? Verify that E[X] = 2 and Var(X) = 2 with the formulas provided at the beginning of this exam.

B (3 Points) We now assume that in the country, there are N couples who can have children, and for a given couple i, we call Xi the random variable corresponding to the number of children this couple will have. Let P be the random variable corresponding to the fraction of boys among all children. Express P in terms of X1, X2,. . . ,XN and N. Then write P in terms of the sample mean XN defined by XN = X1 +X2 +...+XN N C (2 Points) Assume the Xi are independent random variables. What is the expected value of XN and what is the variance of XN? D (4 Points) Assume N is large enough that the answer to the problem is well approximated by taking the limit N → +∞. What does the law of large numbers tell us about X

Answer 1

A)

This is an example of a negative binomial distribution. As per the question, parents will give birth to children until a boy is born and after that they won't reproduce any child anymore.

X = Random variable for the number of child.

As per the question, if X is the number of child then there must be X-1 girls and 1 boy. Hence the probability mass function of X is given by

The mean of negative binomial distribution is given by =

where r = No. of success and p = probability of each success.

hence,

The variance of a negative binomial distribution is given by

B)

Since denotes the random number of children from each couple, hence the total number of children will be

Since, each family stops having child after a boy is born, hence every family will have only 1 boy child. Hence, total number of boys among the children = 1 + 1 + 1 + .......... N times = N

Hence fraction of boys among the children, denoted by P

Now

Putting the above value in the expression for P

we have

C)

Expected value of   

The expected value of X as calculated in the first question is 2.

Hence

Similarly variance of will be calculated in the same manner.

We know that whenever we multiply a certain random variable with some constant then the mean of those random variables get multiplied by the constant and variance of those random variables get multiplied by the square of the constant.

So

D)

The law of large numbers tells us that as more and more couples are taken into consideration, the average value of the random variable X will be closer towards the expected value of X.

Hence, if N is large will be closer and closer to 2.

As N tends to infinity, will become equal to 2.

As N tends to infinity, then the ratio P will be equal to 1/2.

As N tends to infinity,

Hence, as N will tend towards infinity, the ratio P of boys among all the children will become equal to .