Let's solve the problem step-by-step:
1. Identify the Given Information:
- Number of children [tex]\( n = 5 \)[/tex]
- Probability of a child having freckles [tex]\( p = 0.25 \)[/tex]
2. Mean (Expected Value) of [tex]\( X \)[/tex]:
The mean of [tex]\( X \)[/tex] in a binomial distribution [tex]\( X \sim \text{Binomial}(n, p) \)[/tex] is calculated using the formula:
[tex]\[
\mu_X = n \times p
\][/tex]
Substituting the given values:
[tex]\[
\mu_X = 5 \times 0.25 = 1.25
\][/tex]
3. Standard Deviation of [tex]\( X \)[/tex]:
The standard deviation of [tex]\( X \)[/tex] in a binomial distribution [tex]\( X \sim \text{Binomial}(n, p) \)[/tex] is calculated using the formula:
[tex]\[
\sigma_X = \sqrt{n \times p \times (1 - p)}
\][/tex]
Substituting the given values:
[tex]\[
\sigma_X = \sqrt{5 \times 0.25 \times (1 - 0.25)} = \sqrt{5 \times 0.25 \times 0.75}
\][/tex]
Simplifying further:
[tex]\[
\sigma_X = \sqrt{5 \times 0.1875} = \sqrt{0.9375} \approx 0.97
\][/tex]
Thus, the mean [tex]\(\mu_X\)[/tex] is [tex]\(1.25\)[/tex] and the standard deviation [tex]\(\sigma_X\)[/tex] is approximately [tex]\(0.97\)[/tex].
The correct answer from the given options is:
[tex]\[
\mu_X = 1.25, \sigma_X = 0.97
\][/tex]
