math question

Exponential Probabilities

The exponential distribution is often concerned with the amount of time until

some specific event occurs.

For this reason, the exponential distribution is sometimes called the waiting-time

distribution.

You need to rewrite the probabilities in the less than form to use the function in

EXCEL. We will use Excel to find Exponential Probabilities. The probabilities do

need to be in the less than form to use Excel. This is very important.

The Less Than or Equal To Form is not as important because exponential

probabilities are continuous not discrete. But the probabilities do need to be in

the less than form to use Excel.

• P( x = r) • P( x ≤ r) same as P(x < r) • P( x ≥ r) = 1 – P(x < r ) • P(x > r) = 1 – P(x < r) • P(r < x < k) = P(x < k) – P(x < r)

• Expected Value is µ or 𝜆

• For Exponential Distributions using the Excel function you will take 1

𝜆 , and

use this value in the Excel function • r and k are the number of occurrences

To find Exponential Probabilities we will use the =EXPON.DIST( ) function.

To find a certain percentage using Exponential Distribution you will use the

following equation

− ln⁡(1 − 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒)

1 𝜆

You will use the natural log function (ln), the percentage you are looking for and

then 1

𝜆 in the denominator.

Note: this is the same 1

𝜆 that you use in the Excel function.

Example:

A specific species of trees, the western hemlock, was found to have a breast

height diameter distribution that resembled an exponential distribution with

𝜆 = 30.

1) Find the probability that a western hemlock tree growing in the forest has a

diameter that is exactly 23 centimeters in length?

Because of the word “exactly” we want to find this probability P(x = 23). We will

use the EXPON.DIST() function to find this probability.

In Excel you can take =1/30 = .03333. We will use this value in your Excel

function.

P(x = 23) = EXPON.DIST(23,.03333, FALSE)

In Excel make sure you hit the “=“ sign first then start typing in EXPON.DIST( From

here make sure you include the left parenthesis then type in the x value, then

1/lambda, then either TRUE or FALSE. Then close the parenthesis ) and hit Enter.

Type in a TRUE when you have a less than or equal to probability and type in a

FALSE when you have an equals probability. This example has an “=“ sign so we

will use a FALSE.

There is a 1.55% probability that a western hemlock tree growing in the forest has

a diameter that is exactly 23 centimeters in length.

Note: When you hit “Enter” the answer will return as a decimal, .0155. You will

then need to convert it to a percent.

2) Find the probability that a western hemlock tree growing in the forest has a

diameter that is less than 27 centimeters in length?

Because of the word “less than” we will use the less than sign.

This is the probability we want to find, P(x < 27) or P(x ≤ 27)

P(x < 27) = EXPON.DIST(27,.0333,TRUE)

In Excel make sure you hit the “=“ sign first then start typing in EXPON.DIST(.

From here make sure you include the left parenthesis then type in the x value,

then 1/lambda, then either TRUE or FALSE. Then close the parenthesis ) and hit

Enter.

Type in a TRUE when you have a less than probability and type in a FALSE when

you have an equals probability. This example has an “<“ sign so we will use a

TRUE.

There is an 59.34% probability that a western hemlock tree growing in the forest

has a diameter that is less than 27 centimeters in length.

Note: When you hit “Enter” the answer will return as a decimal, .5934. You will

then need to convert it to a percent.

3) Find the probability that a western hemlock tree growing in the forest has a

diameter that is exceeds 25 centimeters in length?

Because of the words “exceeds” we will use the greater than or equal to sign.

This is the probability we want to find, P(x > 25).

This probability is in the greater than or equal to form NOT the less than so we

need to rewrite this in the less than or equal to form.

Remember: P( x > r) = 1 - P( x ≤ r )

P( x > 25) = 1 - P(x ≤ 25). Now that the probability is in the less than form we can

use Excel.

1 - P(x ≤ 25) = 1- EXPON.DIST(25,.0333, TRUE)

In Excel make sure you hit the “=“ sign first, then the 1 - and then, EXPON.DIST(.

From here make sure you include the left parenthesis then type in the x value,

then 1/lambda, then either TRUE or FALSE. Then close the parenthesis ) and hit

Enter.

Type in a TRUE when you have a less than probability and type in a FALSE when

you have an equals probability. This example has an “<“ sign so we will use a

TRUE.

There is a 43.46% probability that a western hemlock tree growing in the forest

has a diameter that is exceeds 25 centimeters in length.

Note: When you hit “Enter” the answer will return as a decimal, .4346. You will

then need to convert it to a percent.

4) Find the probability that a western hemlock tree growing in the forest has a

diameter that is between 28 and 34 centimeters in length?

Because of the word “between” we will this probability P(r < x < k) = P(x < k) – P(x < r) This is the probability we want to find, P(28 < x < 34).

P(28 < x < 34) = P(x < 34) – P(x < 28)

Now that the probability is in the less than form we can use Excel.

P(x < 34) – P(x < 28) = EXPON.DIST(34,.0333,TRUE) – EXPON.DIST(28,.0333,TRUE)

In Excel make sure you hit the “=“ sign first, then EXPON.DIST(. From here make

sure you include the left parenthesis then type in the x value, then 1/lambda,

then TRUE, then close the parenthesis ) and hit the minus sign “ –“ and then

repeat the steps and hit Enter.

Type in a TRUE when you have a less than probability and type in a FALSE when

you have an equals probability. This example has an “<“ sign so we will use a

TRUE.

There is a 7.13% probability that a western hemlock tree growing in the forest has

a diameter that is between 28 and 34 centimeters in length.

Note: When you hit “Enter” the answer will return as a decimal, .0713. You will

then need to convert it to a percent.

63% of all trees will have a diameter of how many centimeters in length?

We will use this equation and plug this equation into Excel:

− ln⁡(1 − 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒)

1 𝜆

There is an LN function in Excel and the percentage we will use is .63

63% of all trees will have a diameter of 29.8276 centimeters.

If you want to plug this in and calculate this by hand. But I find Excel a lot easier.

− ln⁡(1 − 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒)

1 𝜆

− ln⁡(1 − .63)

1 30