The probability density function of the weight of packages delivered by a post office is f(x) = 70/69x^2 for 1 < x < 70 pounds.

a) Determine the mean and variance of weight. Round your answers to two decimal places (e. g. 98.76).
Mean = pounds
Variance = pounds2
b) If the shipping cost is $2.50 per pound, what is the average shipping cost of a package? Round your answer to two decimal places (e. g. 98.76).
pounds
c) Determine the probability that the weight of a package exceeds 59 pounds. Round your answer to four decimal places (e. g. 98.7654).

The weights of the package follows a probability density function

The probability density function is given as:

(a) The mean and the variance

The mean is calculated as:

So, we have:

Rewrite as:

Integrate

Expand

The variance is calculated as:

Where:

So, we have:

Rewrite as:

Integrate

Expand

So, we have:

Hence, the mean is 4.31 and the variance is 51.42, respectively.

(b) The average cost of shipping a package

In (a), we have:

---- Mean

So, the average cost of shipping a package is:

Hence, the average cost of shipping a package is $10.78

(c) The probability a package weighs over 59 pounds

This is represented as: P(x > 59)

So, we have:

So, we have:

Rewrite as:

Integrate

Expand

Hence, the probability a package weighs over 59 pounds is 0.0027

Read more about probability density functions at:

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(a) The mean is 4.31 pounds. The variance is 51.42 pounds.

(b) The average shipping cost of a package is $10.78.

(c) The probability that the weight of a package exceeds 59 pounds is 0.0027.

Step-by-step explanation:

The probability density function of the weight of packages is:

(a)

The formula for expected value (or mean) of X is:

Compute the expected value of X as follows:

Thus, the mean is 4.31 pounds.

The formula to compute the variance is:

Compute the E (X²) as follows:

The variance is:

Thus, the variance is 51.42 pounds.

(b)

It is provided that the shipping cost for per pound is, C = $2.50.

Compute the average shipping cost of a package as follows:

Thus, the average shipping cost of a package is $10.78.

(c)

Compute the probability that the weight of a package exceeds 59 pounds as follows:

Thus, the probability that the weight of a package exceeds 59 pounds is 0.0027.

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