Answer:
To calculate the standard deviation of the distribution, we first need to find the expected value of X, which is the average number of girls in the 8 possible outcomes.
Since each child can be either a boy or a girl with equal probability, there are 2^3 = 8 possible outcomes for the number of boys and girls. We can list these outcomes and the corresponding number of girls:
BBB (0 girls)
BBG (1 girl)
BGB (1 girl)
BGG (2 girls)
GBB (1 girl)
GBG (2 girls)
GGB (2 girls)
GGG (3 girls)
To find the expected value of X, we add up the number of girls in each outcome and divide by the total number of outcomes:
E(X) = (0 + 1 + 1 + 2 + 1 + 2 + 2 + 3) / 8 = 1.5
Next, we need to calculate the variance of X. The variance is the expected value of the squared deviation from the mean:
Var(X) = E[(X - E(X))^2]
We can calculate this by finding the squared deviation from the mean for each possible outcome, multiplying by the probability of that outcome, and summing over all outcomes:
Var(X) = [(0 - 1.5)^2 / 8 + (1 - 1.5)^2 / 4 + (2 - 1.5)^2 / 4 + (3 - 1.5)^2 / 8]
= 0.65625
Finally, the standard deviation is the square root of the variance:
SD(X) = sqrt(Var(X)) = sqrt(0.65625) = 0.81 (rounded to two decimal places)
Interpretation: The standard deviation of the distribution of the number of girls in a family of three children is 0.81. This means that on average, the number of girls in such families is 1.5, but the actual number of girls can vary by up to 0.81 from this average.