Final answer:
The random variable X represents the number of girls in a family with 5 children, where 3 have an equal chance of being girls and there are 2 adopted girls.
The mean of X is 3.5, and the variance is 0.75.
Step-by-step explanation:
The random variable X denotes the number of girls in a family with 5 children, where 3 are natural children with a 50% chance each of being a girl, and 2 are adopted girls.
a) The mean of X, sometimes called the expected value, is calculated by adding the product of each outcome with its probability and taking the sum of these products.
For the natural children, the expected number of girls is 0.5 girls per child.
There are 3 natural children, so the expected number from them is 3 children × 0.5 = 1.5 girls.
Plus the 2 adopted girls always, the mean of X is therefore 3.5 girls.
b) The variance of X measures the spread of the distribution around the mean.
Since the probability of a natural child being a girl is 0.5 (p = 0.5) and being a boy is also 0.5 (1-p = 0.5), the variance for each natural child is p(1-p) = 0.5 × 0.5 = 0.25.
Therefore, for 3 natural children, the variance is 3 children × 0.25 = 0.75.
The adoptive children do not contribute to the variance since their gender is not random. The total variance of X is thus 0.75.