Assume your expected lifetime income is $1.5 million and the standard deviation is $250,000.

a) What is the chance that you will earn more than $1.1 million during your lifetime?

b) What is the chance that you will earn less than $2 million during your lifetime?

c) What is the chance that you will earn between $1.2 and $2.2 million during your lifetime?

d) What is the chance you will make less than $500,000 during your lifetime?

e) With 90% chance your income will be less than how many $$$?

f) With 80% chance your income will be more than how many $$$?

g) With 70% chance your income will be within which symmetric range around $1.5 million?

To answer these questions, we will use the concept of the standard normal distribution.

The standard normal distribution is a probability distribution with a mean of 0 and a standard deviation of 1. We can standardize our lifetime income using the formula:

z = (x - μ) / σ

where z is the standardized score, x is the value of interest, μ is the mean, and σ is the standard deviation.

a) To find the chance of earning more than $1.1 million, we need to find the area under the standard normal curve to the right of the standardized score corresponding to $1.1 million.

1. Standardizing $1.1 million:
z1 = ($1.1 million - $1.5 million) / $250,000

2. Calculating the probability:
P(X > $1.1 million) = P(Z > z1)

From a standard normal distribution table or using a calculator, we can find the probability associated with the standardized score, z1.

b) To find the chance of earning less than $2 million, we need to find the area under the standard normal curve to the left of the standardized score corresponding to $2 million.

1. Standardizing $2 million:
z2 = ($2 million - $1.5 million) / $250,000

2. Calculating the probability:
P(X < $2 million) = P(Z < z2)

Again, from a standard normal distribution table or using a calculator, we can find the probability associated with the standardized score, z2.

c) To find the chance of earning between $1.2 and $2.2 million, we need to find the area under the standard normal curve between the standardized scores corresponding to $1.2 million and $2.2 million.

1. Standardizing $1.2 million:
z3 = ($1.2 million - $1.5 million) / $250,000

2. Standardizing $2.2 million:
z4 = ($2.2 million - $1.5 million) / $250,000

3. Calculating the probability:
P($1.2 million < X < $2.2 million) = P(z3 < Z < z4)

Once again, we can refer to a standard normal distribution table or use a calculator to find the probability between the standardized scores, z3 and z4.

d) To find the chance of earning less than $500,000, we need to find the area under the standard normal curve to the left of the standardized score corresponding to $500,000.

1. Standardizing $500,000:
z5 = ($500,000 - $1.5 million) / $250,000

2. Calculating the probability:
P(X < $500,000) = P(Z < z5)

Using a standard normal distribution table or calculator, we can find the probability associated with the standardized score, z5.

e) To find the income value with a 90% chance of being less, we need to find the standardized score corresponding to a 90th percentile.

1. Finding the z-score:
z6 = invNorm(0.9)

2. Calculating the income value:
90th_Percentile_Income = z6 * $250,000 + $1.5 million

Using the inverse normal distribution function (invNorm) or a standard normal distribution table, we can determine the standardized score associated with the 90th percentile and then convert it back to an income value.

f) To find the income value with an 80% chance of being more, we need to find the standardized score corresponding to a 20th percentile (complement of 80%).

1. Finding the z-score:
z7 = invNorm(0.2)

2. Calculating the income value:
20th_Percentile_Income = z7 * $250,000 + $1.5 million

Using the inverse normal distribution function or a standard normal distribution table, we can find the standardized score associated with the 20th percentile and then convert it back to an income value.

g) To find the symmetric range around $1.5 million with a 70% chance of income falling within it, we need to find the standardized scores corresponding to the desired tails of the distribution.

1. Finding the z-scores for each tail:
z_lower = invNorm((1 - 0.7) / 2)
z_upper = -z_lower

2. Calculating the income range:
Lower_Income = z_lower * $250,000 + $1.5 million
Upper_Income = z_upper * $250,000 + $1.5 million

Using the inverse normal distribution function, we can find the standardized scores corresponding to the desired tails (0.15 each) and then convert them back to income values by multiplying by the standard deviation and adding the mean.

Note: Make sure to use a standard normal distribution table or a calculator with the functions mentioned above to find the standardized scores and calculate the corresponding probabilities or percentile values.