Math Answers

1.A box of billiard balls is inspected. The inspector draws a ball randomly and independently until he finds a ball with a certain brand. It is known that 10% of all balls have that specific brand.

• a). What is the probability distribution function of 𝑋?

• b). What is 𝐸(𝑋) , 𝑉(𝑋) and 𝑀_𝑋(𝑡) ?

• c). What is the probability that 10 balls will be drawn before finding one ball with the required brand?

2.A fair coin is flipped until a head occurs. What is the probability that less than 3 flips are required; that less than 4 flips are required?

1.....Let X be a random variable with a binomial distribution. Then

E(X) = np

V(X) = npq

M_x(t) = (pe^t + q)ⁿ

Using the direct method, prove the above theorems.

2.....9% of students in the class have their bank balances greater than M500. Suppose that 10 students are selected at random to be interviewed about their bank balances

i) What is the probability that two students will have their bank balances greater than M500?

ii) What is the probability that none will have a bank balance greater than M500?

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