8. MINERALS Pyrite, also known as fool's gold, can form crystals that are perfect cubes. Suppose a gemologist wants to cut a cube of pyrite to get a square and a rectanglar face.
What cuts should be made to get each of the shapes? Illustrate your answers.

a)

b)

Step-by-step explanation:

A uniform distribution, sometimes also known as a rectangular distribution, is a distribution that has constant probability and is defined on a interval [a,b].

The Poisson distribution is the discrete probability distribution in order to describe the number of events occurring in a given time period.   And is defined with a parameter calles usually .

Let T the random variable that represent the number of hours between successive train arrivals at the station, and we know that the distribution for T is given by:

The expected value and the variance for an uniform random variable is given by:

ANd let X the random variable that represent the  number of people who get on the next train, so the conditional distribution X|T is given by:

Since we have a Poisson distribution we can find the expected value and the variance for the random variable X|T

We can use this result in order to answer the questions like this:

Part a

Using the properties for expected value we have this:

Part b

We are interested in Var (X) and in order to find this we can use the following property from conditional probability:

And since we already found Var[X|T] and E[X|T] we have this: