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In a family of five children, there is a [tex]$25\%$[/tex] chance that each individual child has freckles. Children inherit freckles independently of one another. Let [tex]$X$[/tex] represent the number of children in this family with freckles. What are the mean and standard deviation of [tex][tex]$X$[/tex][/tex]?

A. [tex]\mu_x = 1.25, \sigma_x = 0.97[/tex]
B. [tex]\mu_x = 1.25, \sigma_x = 0.94[/tex]
C. [tex]\mu_x = 1.5, \sigma_x = 0.19[/tex]
D. [tex]\mu_x = 2.5, \sigma_x = 0.97[/tex]



Answer :

GinnyAnswer
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]

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