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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a. To show that f(x) is a probability density function (pdf), we need to satisfy two conditions:
1. The function must be non-negative for all x.
2. The integral of the function over its entire domain must equal 1.
Let's check these conditions for f(x) = 2x - 3:
1. Non-negativity: For any value of x, f(x) = 2x - 3. Since the coefficient of x (2) is positive, the function is positive for all values of x greater than 1. Therefore, f(x) is non-negative.
2. Integrating over its domain: To find the integral of f(x) over its entire domain, we integrate from 1 to infinity:
∫(1 to ∞) f(x) dx = ∫(1 to ∞) (2x - 3) dx
= [x^2 - 3x] evaluated from 1 to ∞
= (∞^2 - 3∞) - (1^2 - 3*1)
= ∞ - 3∞ - 1 + 3
= -2∞ + 2
The integral is not finite, and thus the condition of having the integral equal to 1 is not satisfied. Therefore, f(x) = 2x - 3 is not a valid probability density function.
b. Since f(x) does not satisfy the conditions to be a pdf, we cannot use it directly to find the mean. However, we can find the mean using an adjusted pdf.
To make f(x) a valid pdf, we need to normalize it by dividing it by an appropriate constant such that the integral becomes 1. Let's find this constant:
∫(1 to ∞) c(2x - 3) dx = c∫(1 to ∞) (2x - 3) dx
= c[x^2 - 3x] evaluated from 1 to ∞
= c[(∞^2 - 3∞) - (1^2 - 3*1)]
= c(-2∞ + 2)
The integral should equal 1, so:
c(-2∞ + 2) = 1
Since (∞ - ∞) is an indeterminate form, we cannot solve this equation directly. Instead, let's calculate the mean in general terms and interpret the result:
The mean (μ) of a random variable X with probability density function f(x) is given by:
μ = ∫(-∞ to ∞) x*f(x) dx
Considering our original function f(x) = 2x - 3 (before normalization), let's calculate the mean for any pdf that satisfies the conditions:
μ = ∫(-∞ to ∞) x(2x - 3) dx
= 2∫(-∞ to ∞) x^2 dx - 3∫(-∞ to ∞) x dx
To find the mean, we need to evaluate these integrals. However, without a specific range or normalization constant, we can't assign a numerical value to the mean.
In summary, the function f(x) = 2x - 3 is not a valid probability density function, and without further information, we cannot determine the mean X.