Question

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

1) What is the random variable for this problem?

2) Confirm that f(x) is a probability density function.

3) Give the cumulative distribution function.

4) Determine the mean.

5) What is the probability that the size of the random particle will be less than 5 micrometers? Be sure to give probability statement.

6)An optical device is being marketed to detect contamination particles. It is capable of detecting particles exceeding 7 micrometers in size. What proportion of the particles will be detected?

7)What is the random variable for this problem?

Answer 1

1)here random variable is the size of  contamination  particle

2)

for above to be valid f(x) dx must be 1

f(x) dx =(2x^-3) dx =(-2x^-2/2)|_{1 ⁼} -0-(-1)=1

therefore f(x) is valid probability density function

3)

CDF =F(x)= f(x) dx =(2x^-3) dx =(-2x^-2/2)|^x_{1 ⁼}1-1/x²

4)

mean E(X)= xf(x) dx =(2x^-2) dx =(-2x^-1/1)|_{1 ⁼} -0-(-2)=2

5)

P(X<5) =F(5)=1-1/5² =1-24/25 =0.96

probability  that the size of the random particle will be less than 5 micrometers is 0.96.

6)

proportion of the particles will be detected =P(X>7) =1-P(X<7) =1-(1-1/7²) =1/49 =0.0204

7)

random variable is proportion of  particles that will be detected by optical device