Statistics and Probability Answers

1. Let X be a random variable having a Bernoulli distribution. Then

• a) E[X] = p

• b) V[X] = pq = p(1-p)

• c) M_X(t) = pe^t + q

• Prove the above equalities.

2.Let 𝑋 be a Bernoulli probability mass function with

M_X(t) = pe^t + q

Derive the mean and variance of 𝑋.

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}\n \\frac{x-10}{1800} ,& 10<x<70 \\\\\n 0, & otherwise\n\\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}\n 3[3-16(x-10)^2], & 9.75 <x<10.25 \\\\\n 0, & otherwise\n\\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.

a) Given a probability distribution of some discrete distribution as

p_x(x) = "\\theta"^x (1 - "\\theta") ^{1 - x}, 0 "\\leq x \\leq 1"

• Find the moment generating function of X.

b) Given a probability distribution of some continuous distribution as

f_x(x) = "\\begin{Bmatrix}\n \\frac {1}{10}, & 20 \\leq x \\leq 30 \\\\\n 0, & elsewhere\n\\end{Bmatrix}"

• Find the moment generating function of X.

Suppose that the only 2 possible values of a random variable 𝑋are 0 and 1.

• Let 𝑃[𝑋 = 0] = 0.1

• 𝑃[𝑋 = 1] = 0.9

• Find the k^th moment of the random variable 𝑋.

• Let x₁ = −1, x₂ = 0, x₃ = 1, x₄ = 2.

• Find the first 4 central moments

a)  Define R_x = {1, 2,3, 4} and a function g(x) as follows:

g(x) =

Is the above a probability function of some random variable?