1) 17/10
2)41/10
3) b/a
There are 100 families: 10 families have no children, 40 families have 1 child each, 30 families have 2 children each, 10 families have 3 each, and 10 families have 4 each.
If you pick a family at random (each family in the village being equally likely to be picked), what is the expected number of children in that family?
- unanswered
If you pick a child at random (each child in the village being equally likely to be picked), what is the expected number of children in that child's family (including the picked child)?
- unanswered
Generalize your approach from part 2: Suppose that a fraction pk of the families have k children each. Let K be the number of children in a randomly selected family, and let a=E[K] and b=E[K2]. Let W be the number of children in a randomly chosen child's family. Express E[W] in terms of a and b using standard notation.
E[W]=
- unanswered
2)41/10
3) b/a
41/17
For the given statistics, we have:
- 10 families with no children (0 children each)
- 40 families with 1 child each
- 30 families with 2 children each
- 10 families with 3 children each
- 10 families with 4 children each
To calculate the expected number of children in a family, we multiply the number of children in each category by the probability of selecting a family from that category and sum them up.
Expected number of children = (0 * 10/100) + (1 * 40/100) + (2 * 30/100) + (3 * 10/100) + (4 * 10/100)
= (0 + 40/100 + 60/100 + 30/100 + 40/100)
= (0 + 0.4 + 0.6 + 0.3 + 0.4)
= 2.1
Therefore, the expected number of children in a randomly selected family is 2.1.
Now, let's calculate the expected number of children in a child's family. Since each child is equally likely to be picked, we need to consider the probability of a child belonging to a family with a certain number of children.
To find this, we divide the number of families with a certain number of children by the total number of families. Let's denote the number of children in a randomly selected child's family as W.
For W = 0:
Probability(W = 0) = Number of families with 0 children / Total number of families = 10/100 = 0.1
For W = 1:
Probability(W = 1) = Number of families with 1 child / Total number of families = 40/100 = 0.4
Similarly, we can calculate the probabilities for W = 2, 3, and 4.
Expected number of children in a child's family (E[W]) = (0 * 0.1) + (1 * 0.4) + (2 * 0.3) + (3 * 0.1) + (4 * 0.1)
= (0 + 0.4 + 0.6 + 0.3 + 0.4)
= 2.1
So, the expected number of children in a randomly selected child's family is also 2.1.
Now, let's generalize this approach for any given fraction pk of families having k children each and calculate the expected number of children in a randomly chosen child's family using the variables a and b.
E[W] = (0 * p0) + (1 * p1) + (2 * p2) + ... + (k * pk)
Note: Here, p0 + p1 + p2 + ... + pk = 1 (because the sum of probabilities should be equal to 1).
Therefore, E[W] = p1 + 2p2 + 3p3 + ... + kpk
This notation is commonly used to express the expected value of a random variable.