Question

The random variable x represents the number of boys in a family with three children. Assuming that births of boys and girls are equally likely, find the mean and standard deviation for the random variable x.

Answer 1

Answer:

$X \sim Binom(n=3, p=0.5)$

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

Th expected value is given by:

$E(X) = np= 3*0.5 = 1.5$

The variance is given by:

$Var(X) = 3*0.5*(1-0.5) =0.75$

And the deviation is given by:

$\sigma =\sqrt{0.75}=0.866$

Step-by-step explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest "number of boys in a family", on this case we now that:

$X \sim Binom(n=3, p=0.5)$

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

Th expected value is given by:

$E(X) = np= 3*0.5 = 1.5$

The variance is given by:

$Var(X) = 3*0.5*(1-0.5) =0.75$

And the deviation is given by:

$\sigma =\sqrt{0.75}=0.866$