5) You know that you will inherit 20% of your uncle John’s wealth, 30% of your aunt Becky’s wealth and 50% of your parent’s wealth. Since they are not dead yet, their respective wealth is a RV and all we know is that their wealth at death is ~N and independent of each other. John’s wealth is J~N(1000,1002), Becky’s is B~N(2000,3002) and your parents’ wealth is P~N(5000,5002).
a) What is the distribution of you total inheritance?
b) What is the chance you will inherit at least $3000?
c) You are 40% sure to inherit at most how much?
a) To find the distribution of your total inheritance, we need to find the distribution of the sum of the inheritances from each individual.
Let's calculate the mean and variance of each inheritance first:
For your uncle John's wealth (J), the mean is 1000 and the variance is 1002.
For your aunt Becky's wealth (B), the mean is 2000 and the variance is 3002.
For your parents' wealth (P), the mean is 5000 and the variance is 5002.
Now, let's find the distribution of your total inheritance (T), which is the sum of 20% of J, 30% of B, and 50% of P.
Mean of T = 0.2 * 1000 + 0.3 * 2000 + 0.5 * 5000 = 500 + 600 + 2500 = 3600
Variance of T = (0.2^2) * 1002 + (0.3^2) * 3002 + (0.5^2) * 5002 = 40.16 + 270.18 + 1250.5 = 1560.84
Therefore, the distribution of your total inheritance T is T ~ N(3600, 1560.84).
b) To find the chance you will inherit at least $3000, we need to calculate the probability that your total inheritance T is greater than or equal to $3000.
P(T ≥ 3000) = P(Z ≥ (3000 - 3600) / sqrt(1560.84))
Here, Z is the standard normal distribution.
We can use a standard normal distribution table (or software) to find the probability associated with the Z-value calculated above.
c) To find the maximum amount you inherit with 40% certainty, we need to find the value of X such that P(T ≤ X) = 0.4.
This involves finding the Z-value corresponding to a cumulative probability of 0.4 and then using the mean and standard deviation of the total inheritance distribution to calculate X.
P(T ≤ X) = 0.4
P(Z ≤ (X - 3600) / sqrt(1560.84)) = 0.4
We can then find the corresponding Z-value using the standard normal distribution table or software, and solve for X using the formula:
X = sqrt(1560.84) * Z + 3600
a) To find the distribution of your total inheritance, we need to consider the probability distribution of each individual's wealth and then calculate the corresponding probabilities.
Let X represent your total inheritance. Since the wealth of your uncle John, aunt Becky, and your parents are independent, the distribution of your total inheritance can be found by taking the sum of the three distributions, i.e., X = J + B + P.
Given:
J ~ N(1000, 1002)
B ~ N(2000, 3002)
P ~ N(5000, 5002)
The mean and variance of the sum of independent random variables can be calculated as follows:
E[X] = E[J] + E[B] + E[P]
Var(X) = Var(J) + Var(B) + Var(P)
Using the above formulas, the mean and variance of your total inheritance can be calculated as:
E[X] = 1000 + 2000 + 5000 = 8000
Var(X) = 1002 + 3002 + 5002 = 9006
Therefore, the distribution of your total inheritance is X ~ N(8000, 9006).
b) To find the chance you will inherit at least $3000, we need to calculate the probability P(X >= 3000).
Using the properties of the normal distribution, we can standardize the variable to Z-score and then lookup the probability in the standard normal table. Let Z represent the Z-score:
Z = (X - mean) / standard deviation
Z = (3000 - 8000) / sqrt(9006)
Z ≈ -24.99
Looking up the Z-score in the standard normal table, we find that the probability is extremely close to 0, indicating that the chance you will inherit at least $3000 is almost negligible.
c) To calculate the maximum inheritance with 40% certainty, we need to find the corresponding value of X that has a cumulative probability of 0.4 (40%).
Using the Z-score formula, let's solve for X:
Z = (X - mean) / standard deviation
0.4 = (X - 8000) / sqrt(9006)
Rearranging the equation to solve for X:
X - 8000 = 0.4 * sqrt(9006)
X ≈ 8000 + (0.4 * sqrt(9006))
Calculating this value, we find that you are 40% sure to inherit at most approximately 8512.