An article in the European Physical Journal B ["Evidence for the exponential distribution of income in the USA" (2001, Vol. 20, pp. 585-589)] shows that individual income in the US can be approximated with an exponential distribution. Suppose that the mean of individual income in a year is $23000. (a) Write the probability distribution function for individual income. f (x) = Edit LINK TO TEXT (b) What percentage of individuals in the US earn less than $21000? Round your answer to three decimal places (e.9. 98.765). P(X < 21000) = L (c) What percentage of individuals in the US earn more than $495007 Round your answer to three decimal places (e. g. 98.765). P(X > 49500) = (d) What is the income exceeded by 2% of individuals?

a) f(x) = P(X=x) = λ e^(-λ.x)

b) P(X < 21000) = 0.59868 = 59.868%

c) P(X > 49500) = 0.11623 = 11.623%

d) $464.66

Step-by-step explanation:

a) Exponential random variable's probability function is given as

f(x) = P(X=x) = λ e^(-λ.x)

The cumulative distribution function is given as

P(X < x) = 1 - e^(-λ.x)

where λ = rate parameter = (1/mean)

b) The percentage of individuals in the US earn less than $21000

P(X < 21000)

P(X < x) = 1 - e^(-λ.x)

x = 21000

λ = (1/23000) = 0.0000434783

λ × x = 0.0000434783 × 21000 = 0.913

P(X < 21000) = 1 - e⁻⁰•⁹¹³ = 0.59868 = 59.868%

c) The percentage of individuals in the US earn more than $49500

P(X > 49500)

P(X > x) = 1 - [1 - e^(-λ.x)] = e^(-λ.x)

x = 49500

λ = (1/23000) = 0.0000434783

λ × x = 0.0000434783 × 49500 = 2.152

P(X > 49500) = e⁻²•¹⁵² = 0.1162312064 = 0.11623 = 11.623%

d) What is the income exceeded by 2% of individuals?

P(X < x) = 2% = 0.02

P(X |< x) = 1 - e^(-λ.x) = 0.02

e^(-λ.x) = 0.98

In [e^(-λ.x)] = In 0.98 = -0.0202027073

-λ.x = -0.0202027073

x = 0.0202027073 ÷ λ

x = 0.0202027073 ÷ 0.0000434783

x = $464.66

Hope this Helps!!!

2 & 7

step-by-step explanation:

he climbs 300 ft/hr at a constant rate. the peak is at 1,500 ft.

when going up, he reaches 600 ft in 2 hours. remember, that he is going in a constant rate. divide 600 with 300:

600/300 = 2

assuming that he takes the same route back (meaning that it's a 1500 ft route back, you can solve.

note also that you are solving when he gets back to 600 ft after he reaches the peak.

first, find the amount of hours it takes to get up the peak. divide the total amount (1500), with the ft/her (300) to get the amount of hours (x):

1500/300 = x

x = 5

it takes jamie 5 hours to get up. now, he is returning down, and reaches 600 (his current destination). remember that he climbs at a constant 300 ft/hr. divide 600 with 300:

600/300 = x

x = 2

now add the two numbers together.

x = 5 + 2 = 7

jamie takes 7 hours to go up the mountain and reach his current position when descending.

(total amount/constant rate) = (time).

remember that it is a constant rate, so in total, when he gets back down the mountain, it will take the same amount of time as when he goes up. the total amount for the round trip is: 5 + 5 = 10.

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a

step-by-step explanation: