Particles are a major component of air pollution in many areas. It is of interest to study the sizes of contaminating particles. Let X represent the diameter, in micrometers, of a randomly chosen particle. Assume that in a certain area, the probability density function of X is inversely proportional to the volume of the particle; that is, assume that
where c is a constant.
c/x^4 x > or equal to 1
0 > 1
(a) Find the value of c so that is a valid probability density function.
(b) Find the mean particle diameter.
(c) Find the median particle diameter.
(d) The term refers to particles 10 or less in diameter. What proportion of the contaminating particles are ?
(e) The term refers to particles 2.5 or less in diameter. What proportion of the contaminating particles are ?
(f) What proportion of the particles are ?
To find the value of c, we need to make sure that the probability density function (pdf) is valid, which means it integrates to 1 over its domain.
(a) To find c, we integrate the pdf from 1 to infinity and set it equal to 1:
∫(c/x^4)dx from 1 to ∞ = 1
To integrate c/x^4, we can rewrite it as c * x^(-4):
c ∫x^(-4)dx from 1 to ∞ = 1
To integrate x^(-4), we add 1 to the power and divide by the new power:
c * [(x^(-3))/(-3)] from 1 to ∞ = 1
Applying the limits of integration:
c * [0 - (1^(-3))/(-3)] = 1
Simplifying:
c * (1/3) = 1
c = 3
So, the value of c is 3.
(b) The mean or expected value of a continuous random variable can be found by integrating the product of the variable and the pdf over its domain.
To find the mean particle diameter (mean or expected value of X), we calculate:
E[X] = ∫(x * (3/x^4))dx from 1 to ∞
Simplifying:
E[X] = 3 ∫x^(-3)dx from 1 to ∞
Integrating x^(-3):
E[X] = 3 * [(x^(-2))/(-2)] from 1 to ∞
Applying the limits of integration:
E[X] = 3 * [0 - (1^(-2))/(-2)]
Simplifying:
E[X] = 3 * [0 - (-1/2)]
E[X] = 3 * (1/2)
E[X] = 3/2
So, the mean particle diameter is 3/2 micrometers.
(c) The median is the value that splits the distribution into two equal halves. To find the median particle diameter, we need to find the value of X for which the cumulative distribution function (CDF) is equal to 0.5.
CDF(x) = ∫(3/x^4)dx from 1 to x
Setting CDF(x) = 0.5 and solving for x:
0.5 = ∫(3/t^4)dt from 1 to x
0.5 = 3 * ∫t^(-4)dt from 1 to x
0.5 = 3 * [(t^(-3))/(-3)] from 1 to x
0.5 = -1 * (x^(-3))/3 - (-1 * (1^(-3))/3)
0.5 = -(x^(-3))/3 + 1/3
0.5 = 1/3 - (x^(-3))/3
Rearranging the equation:
(x^(-3))/3 = 1/3 - 0.5
(x^(-3))/3 = 1/3 - 1/2
(x^(-3))/3 = (2 - 3)/6
(x^(-3))/3 = -1/6
Dividing both sides by (1/6):
(x^(-3))/(3 * (1/6)) = -1
2 * (x^(-3)) = -1
(x^(-3)) = -1/2
Taking the reciprocal of both sides:
x^3 = -2
Since we cannot take the cube root of a negative number, there is no real solution for x. This means that the median particle diameter cannot be directly calculated from the given probability density function.
(d) The term P(X ≤ 10) refers to the proportion of particles with a diameter of 10 micrometers or less. To find this proportion, we calculate the cumulative distribution function (CDF) at x = 10.
CDF(10) = ∫(3/x^4)dx from 1 to 10
CDF(10) = ∫(3/x^4)dx from 1 to 10
CDF(10) = 3 ∫(1/t^4)dt from 1 to 10
CDF(10) = 3 * [(t^(-3))/(-3)] from 1 to 10
CDF(10) = -1/3 * (10^(-3)) - (-1/3 * (1^(-3)))
CDF(10) = -1/3 * (1/1000) - (-1/3 * 1)
CDF(10) = -1/3000 + 1/3
CDF(10) = (1 - 1/3000) / 3
CDF(10) = (3000 - 1) / 3000 / 3
CDF(10) = 2999 / 3000 / 3
CDF(10) = 0.9996666667
So, approximately 99.97% of the contaminating particles have a diameter of 10 micrometers or less.
(e) The term P(X ≤ 2.5) refers to the proportion of particles with a diameter of 2.5 micrometers or less. To find this proportion, we calculate the cumulative distribution function (CDF) at x = 2.5 using the same approach as in part (d).
CDF(2.5) = 3 ∫(1/t^4)dt from 1 to 2.5
CDF(2.5) = -1/3 * (2.5^(-3)) - (-1/3 * (1^(-3)))
CDF(2.5) = -1/3 * (1/2.5^3) + 1/3
CDF(2.5) = -1/3 * (1/15.625) + 1/3
CDF(2.5) = -1/3 * (0.064) + 1/3
CDF(2.5) = -0.0213333333 + 1/3
CDF(2.5) = 0.9786666667
So, approximately 97.87% of the contaminating particles have a diameter of 2.5 micrometers or less.
(f) The term P(2 ≤ X ≤ 3) refers to the proportion of particles with a diameter between 2 and 3 micrometers. To find this proportion, we calculate the difference between the cumulative distribution function (CDF) at x = 3 and x = 2.
P(2 ≤ X ≤ 3) = CDF(3) - CDF(2)
P(2 ≤ X ≤ 3) = ∫(3/x^4)dx from 1 to 3 - ∫(3/x^4)dx from 1 to 2
Using the same integration and simplification as before, we can calculate the exact proportion.