Question

in a family of five children, there is a 25% chance that each individual child has freckles. children inherit freckles independently of one another. let x represent the number of children in this family with freckles. what are the mean and standard deviation of x?
$mu _{x}=1.25,sigma _{x}=0.97$
$mu _{x}=1.25,sigma _{x}=0.94$
$mu _{x}=1.5,sigma _{x}=0.19$
$mu _{x}=2.5,sigma _{x}=0.97$

Explanation:

Step1: Identify the distribution

This is a binomial distribution problem. The binomial distribution formula for the mean $\mu = np$ and standard deviation $\sigma=\sqrt{np(1 - p)}$, where $n$ is the number of trials and $p$ is the probability of success in a single trial. Here, $n = 5$ (number of children) and $p=0.25$ (probability of a child having freckles).

Step2: Calculate the mean

Using the formula $\mu=np$, substitute $n = 5$ and $p = 0.25$.
$$\mu=5\times0.25=1.25$$

Step3: Calculate the standard deviation

First, find $1 - p=1 - 0.25 = 0.75$. Then use the formula $\sigma=\sqrt{np(1 - p)}$. Substitute $n = 5$, $p = 0.25$, and $1 - p=0.75$.
$$\sigma=\sqrt{5\times0.25\times0.75}=\sqrt{0.9375}\approx0.97$$

Answer:

$\mu_{x}=1.25,\sigma_{x}=0.97$ (the first option)