Suppose that contamination particle size in micrometers can be modeled as f(x) = 2x−3 for 1 < x.

a. Show that f(x) is a probability density function.
b. Find the mean X.

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To show that f(x) = 2x - 3 is a probability density function (pdf), we need to satisfy two conditions:

1. The function must be non-negative for all x:
We have f(x) = 2x - 3, and since x > 1, f(x) = 2x - 3 > 0 for all x > 1.

2. The total area under the curve must be equal to 1:
To find the total area, we need to find the definite integral of f(x) over its entire range:
∫(1 to ∞) f(x) dx = ∫(1 to ∞) (2x - 3) dx

Performing the integration, we get:
∫(1 to ∞) (2x - 3) dx = [x^2 - 3x] evaluated from 1 to ∞
As x approaches infinity, the value becomes infinite, so we have an improper integral.
To calculate it, we can take the limit as a approaches infinity:
lim[a→∞] of ∫(1 to a) (2x - 3) dx

Evaluating the definite integral, we get:
[x^2 - 3x] evaluated from 1 to a = (a^2 - 3a) - (1^2 - 3(1)) = a^2 - 3a - 1

Taking the limit as a approaches infinity:
lim[a→∞] of (a^2 - 3a - 1) = ∞

Since the limit is infinite, the total area under the curve is infinite. However, we can normalize the pdf by dividing the function by the total area.

To find the mean X, we need to calculate the expected value of the random variable X, which corresponds to the mean of the pdf.

The expected value, E(X), can be found using the formula:
E(X) = ∫(1 to ∞) x * f(x) dx

Substituting f(x) = 2x - 3 into the formula, we get:
E(X) = ∫(1 to ∞) x * (2x - 3) dx

Performing the integration, we get:
E(X) = ∫(1 to ∞) (2x^2 - 3x) dx = [2/3 * x^3 - (3/2) * x^2] evaluated from 1 to ∞

As x approaches infinity, the value becomes infinite, so we have another improper integral.
To calculate it, we can take the limit as a approaches infinity:
lim[a→∞] of ∫(1 to a) (2x^2 - 3x) dx

Evaluating the definite integral, we get:
[2/3 * x^3 - (3/2) * x^2] evaluated from 1 to a = (2/3 * a^3 - (3/2) * a^2) - (2/3 * 1^3 - (3/2) * 1^2)
Simplifying:
E(X) = (2/3 * a^3 - (3/2) * a^2) - (-1/6) = 2/3 * a^3 - (3/2) * a^2 + 1/6

Taking the limit as a approaches infinity:
lim[a→∞] of (2/3 * a^3 - (3/2) * a^2 + 1/6) = ∞

Again, since the limit is infinite, the mean of X is also infinite. This means that the particles' contamination size has no finite mean value.