Question

1. The hydrogenation of benzene to cyclohexane is promoted with a finely divided porous nickelcatalyst.

The catalyst particles can be consideredtobespheresofvarioussizes.Allthe particleshave masses between 10 𝑎nd 70 𝜇g. Let 𝑋 be themassofarandomly chosen particle.The probability density function of 𝑋 is given by

f(x) = $\begin{Bmatrix} \frac{x-10}{1800} ,& 10<x<70 \\ 0, & otherwise \end{Bmatrix}$

A) What proportion of particles have masses less than 50𝜇g?

B) Find the mean mass of the particles.

C) Find the standard deviation of the particles masses.

Find the cumulative distribution function of the particle masses.

2. A process that manufactures piston rings produces rings whose diameters (in centimeters) varyaccording to the probability density function

f(x) = $\begin{Bmatrix} 3[3-16(x-10)^2], & 9.75 <x<10.25 \\ 0, & otherwise \end{Bmatrix}$

a)Is the above function a probability density function?

b) find the mean and standard deviation of diameter of rings manufactured by this process.

Answer 1

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