PROBLEM 7: SAMPLING FAMILIES (3 points possible)
We are given the following statistics about the number of children in the families of a small village.
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
17/10
41/17
b/a
Well, well, well... let's see what we have here, a question about families and children. It's like a little census party! So, they've given us some statistics about the number of children in the families of a small village. Now, the question is asking us to find the expected number of children in a randomly picked family AND the expected number of children in a randomly picked child's family. That seems quite fascinating!
Now, for the first part of the question, we need to find the expected number of children in a randomly picked family. Easy peasy, lemon squeezy! We just need to calculate the average. Let's break it down. We have 10 families with no children, 40 families with 1 child, 30 families with 2 children, 10 families with 3 children, and 10 families with 4 children. So, if we calculate the expected number of children in a randomly picked family, it would be:
(0 children * 10 families) + (1 child * 40 families) + (2 children * 30 families) + (3 children * 10 families) + (4 children * 10 families)
-------------------------------------------------------
100 total families
That leads us to the exciting answer of... drumroll, please... 1.7! Yes, that's right, on average, there are 1.7 children in a randomly picked family in the village. Isn't that...oddly specific?
Now, moving on to the second part of the question. We need to find the expected number of children in a randomly picked child's family, including the picked child. Well, well, well... this seems like a tough one. But hey, tough questions are my bread and butter! Let's break it down again. We have a randomly picked child, and we want to find the expected number of children in their family. Hmmm... I'm gonna need a moment to think about this one. *insert awkward silence*
Ah-ha! I've got it! To find the expected number of children in a randomly picked child's family, we need to go down a wibbly-wobbly path. We already know that on average, there are 1.7 children in a randomly picked family. So, if we include the picked child, we just add one to that average. So, the expected number of children in a randomly picked child's family is... 2.7! Wow, that's a little more than the average number of children in a randomly picked family, isn't it? Kids these days, always trying to outdo their parents!
And, finally, for the generalization part. We have the number of children in a randomly selected family, denoted as K. The expected value of K is represented as E[K]. Similarly, we have the number of children in a randomly selected child's family, denoted as W. The expected value of W is represented as E[W]. Now, we want to express E[W] in terms of a, which is E[K], and b, which is E[K^2]. Ah, we're getting fancy with the notation!
To express E[W] in terms of a and b, well, I hate to disappoint you, but that's still unanswered. Ohhhh, the suspense is killing me! But fear not, my friend, I'm sure you'll find the answer in your study materials. Just remember, when it comes to expectations and averages, it's all about calculating the average values and adding or subtracting a little twist here and there.
Good luck on your journey to find the elusive E[W], my curious friend! May the mathematical gods be with you.
To calculate the expected number of children in a family, we can use the concept of expected value.
In this problem, we are given the statistics about the number of children in the families. We can represent this information in a table as follows:
Number of children (k) Number of families (f)
0 10
1 40
2 30
3 10
4 10
To find the expected number of children in a family, we need to calculate the weighted average of the number of children in each category.
The formula for calculating the expected value is:
E[x] = Σ(x * P(x))
Where x represents the number of children and P(x) represents the probability of having x children in a family.
Using this formula, we can calculate the expected number of children in a family as follows:
E[x] = (0 * P(0)) + (1 * P(1)) + (2 * P(2)) + (3 * P(3)) + (4 * P(4))
Given that there are 100 families in total, we can calculate the probabilities as follows:
P(0) = 10/100 = 0.1
P(1) = 40/100 = 0.4
P(2) = 30/100 = 0.3
P(3) = 10/100 = 0.1
P(4) = 10/100 = 0.1
Substituting these probabilities into the equation, we get:
E[x] = (0 * 0.1) + (1 * 0.4) + (2 * 0.3) + (3 * 0.1) + (4 * 0.1)
= 0 + 0.4 + 0.6 + 0.3 + 0.4
= 2.7
Therefore, the expected number of children in a family is 2.7.
Now, let's move on to the second part of the problem.
To find the expected number of children in a child's family (including the picked child), we need to consider all possible families that the child could belong to.
From the given statistics, we can see that a child can belong to a family with 0, 1, 2, 3, or 4 children.
Using the same approach as before, we can calculate the expected number of children in a child's family as follows:
E[W] = (0 * P(0)) + (1 * P(1)) + (2 * P(2)) + (3 * P(3)) + (4 * P(4))
Using the probabilities calculated earlier, we get:
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
= 2.7
Therefore, the expected number of children in a child's family (including the picked child) is also 2.7.
Now let's move on to the generalized approach in part 3.
In the generalized scenario, let pk be the fraction of families that have k children each.
To calculate E[W], the expected number of children in a child's family, we can use the values of a (E[K]) and b (E[K^2]).
E[W] can be expressed as:
E[W] = a + 1
This is because a represents the expected number of children in a family, and adding 1 accounts for the picked child.
Using standard notation, we can conclude that E[W] = a + 1.
I hope this helps! Let me know if you have any further questions.
To find the expected number of children in a family, we need to calculate the mean of the number of children in each family. This can be done by multiplying the number of children in each family by the probability of selecting that family, and then summing up these values.
In this case, we are given the statistics about the number of children in each family in the village. Let's calculate the expected number of children in a family.
We have the following data:
- 10 families have no children (0 children)
- 40 families have 1 child each
- 30 families have 2 children each
- 10 families have 3 children each
- 10 families have 4 children each
To calculate the expected number of children, we multiply the number of children in each family by the probability of selecting that family, and then sum up these values.
The probability of selecting each family can be calculated by dividing the number of families with a certain number of children by the total number of families. In this case, the total number of families is 100.
So, the probability of selecting a family with no children is 10/100 = 0.1.
The probability of selecting a family with 1 child is 40/100 = 0.4.
The probability of selecting a family with 2 children is 30/100 = 0.3.
The probability of selecting a family with 3 children is 10/100 = 0.1.
The probability of selecting a family with 4 children is 10/100 = 0.1.
Now, we can calculate the expected number of children:
E[number of children] = (0 * 0.1) + (1 * 0.4) + (2 * 0.3) + (3 * 0.1) + (4 * 0.1)
= 0 + 0.4 + 0.6 + 0.3 + 0.4
= 1.7
Therefore, the expected number of children in a randomly selected family is 1.7.
Now let's move on to the second part of the question.
To find the expected number of children in a child's family (including the picked child), we need to calculate the mean of the number of children in each family, considering the probability of selecting a child.
Using the statistics given, we can calculate the probability of selecting a child from each family by dividing the number of children in each family by the total number of children in the village.
The total number of children in the village can be calculated by summing up the number of children in each family.
Total number of children = (0 * 10) + (1 * 40) + (2 * 30) + (3 * 10) + (4 * 10)
= 0 + 40 + 60 + 30 + 40
= 170
Now, the probability of selecting a child from each family can be calculated by dividing the number of children in each family by the total number of children.
The probability of selecting a child from a family with no children is 0/170 = 0.
The probability of selecting a child from a family with 1 child is 1/170 = 0.00588.
The probability of selecting a child from a family with 2 children is 2/170 = 0.01176.
The probability of selecting a child from a family with 3 children is 3/170 = 0.01765.
The probability of selecting a child from a family with 4 children is 4/170 = 0.02353.
Now, we can calculate the expected number of children in a child's family:
E[number of children in child's family] = (0 * 0) + (1 * 0.00588) + (2 * 0.01176) + (3 * 0.01765) + (4 * 0.02353)
= 0 + 0.00588 + 0.02352 + 0.05295 + 0.09412
= 0.17647
Therefore, the expected number of children in a randomly selected child's family (including the picked child) is approximately 0.17647.
For the third part of the question, we are asked to generalize the approach. Let's denote pk as the fraction of families with k children each. We are given that K is the number of children in a randomly selected family, and a=E[K] and b=E[K2].
To express the expected number of children in a randomly chosen child's family (W) in terms of a and b, we can use the variance formula:
Var[X] = E[X2] - (E[X])2
Here, Var[W] = E[W2] - (E[W])2.
Since W is the number of children in the child's family, we can express it as W = K + 1.
Now, let's calculate E[W]:
E[W] = E[K + 1]
= E[K] + E[1]
= E[K] + 1
Since a=E[K], we can substitute it into the equation:
E[W] = a + 1
Therefore, the expected number of children in a randomly chosen child's family (W) is a + 1.
I hope this explanation helps! Let me know if you have any further questions.