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.

This is a binomial distribution with parameters

n=3 (number of trials in the experiment)
p=1/2 (probability of boys, X)
q=1-p (probability of girls)

For a random variable X with a binomial distribution (n,p,q), then
E(x)=μ=np
σ²=npq
where σ=standard deviation (of population)

To find the mean and standard deviation for the random variable x representing the number of boys in a family with three children, we need to first determine the possible outcomes.

The three children's genders can be represented by three-letter combinations using "B" for a boy and "G" for a girl. The possible outcomes are:

BBB (3 boys)
BBG, BGB, GBB (2 boys)
BGG, GBG, GGB (1 boy)
GGG (0 boys)

Now, let's assign probabilities to these outcomes. Since it is given that births of boys and girls are equally likely, we can assume that each of these outcomes has a 1/8 (or 0.125) probability.

Next, we calculate the mean:

Mean (μ) = (Value1 × Probability1) + (Value2 × Probability2) + ... + (ValueN × ProbabilityN),

where Value1, Value2, ..., ValueN are the possible values of the random variable (in this case, the number of boys) and Probability1, Probability2, ..., ProbabilityN are their corresponding probabilities.

Using this formula, we get:

Mean = (3 × 1/8) + (2 × 3/8) + (1 × 3/8) + (0 × 1/8)
= 3/8 + 6/8 + 3/8 + 0
= 12/8
= 1.5

Therefore, the mean of the random variable x is 1.5.

Now, let's calculate the standard deviation:

Standard Deviation (σ) = sqrt((Value1 - Mean)^2 × Probability1 + (Value2 - Mean)^2 × Probability2 + ... + (ValueN - Mean)^2 × ProbabilityN)

Using this formula, we get:

Standard Deviation = sqrt((3 - 1.5)^2 × 1/8 + (2 - 1.5)^2 × 3/8 + (1 - 1.5)^2 × 3/8 + (0 - 1.5)^2 × 1/8)
= sqrt(1.125 + 0.375 + 0.375 + 1.125)
= sqrt(3)

Therefore, the standard deviation of the random variable x is sqrt(3).

To find the mean and standard deviation for the random variable x, we need to calculate the expected value and variance.

1. Mean (Expected Value):
The mean (μ) of a random variable is the sum of all possible values of the variable multiplied by their respective probabilities. In this case, the possible values for x are 0, 1, 2, and 3.

The probability of having 0 boys is (1/2) * (1/2) * (1/2) = 1/8, as there is a 1/2 chance of having a girl and a 1/8 chance of having no boys.
The probability of having 1 boy is (1/2) * (1/2) * (1/2) = 1/8, as each child has a 1/2 chance of being a boy.
The probability of having 2 boys is (1/2) * (1/2) * (1/2) = 1/8, following the same logic.
The probability of having 3 boys is (1/2) * (1/2) * (1/2) = 1/8.

Now we can calculate the mean:
Mean (μ) = (0 * 1/8) + (1 * 1/8) + (2 * 1/8) + (3 * 1/8)
Mean (μ) = 0/8 + 1/8 + 2/8 + 3/8
Mean (μ) = 6/8
Mean (μ) = 3/4

So, the mean of the random variable x is 3/4.

2. Standard Deviation:
To find the standard deviation (σ), we need to calculate the variance first.

Variance (Var) = (0 - μ)^2 * (1/8) + (1 - μ)^2 * (1/8) + (2 - μ)^2 * (1/8) + (3 - μ)^2 * (1/8)

Calculate each term:
(0 - 3/4)^2 * (1/8) = 9/64 * 1/8 = 9/512
(1 - 3/4)^2 * (1/8) = 1/16 * 1/8 = 1/128
(2 - 3/4)^2 * (1/8) = 1/16 * 1/8 = 1/128
(3 - 3/4)^2 * (1/8) = 9/64 * 1/8 = 9/512

Now we can calculate the variance:
Variance (Var) = 9/512 + 1/128 + 1/128 + 9/512
Variance (Var) = 21/512

Finally, calculate the standard deviation (σ):
Standard Deviation (σ) = √(Variance)
Standard Deviation (σ) = √(21/512)

Therefore, the standard deviation for the random variable x is √(21/512).