Sample question 1: Statistical knowledge

Suppose there are 15 different color crayons in a box. Each time one obtains a crayon, it is equally likely to be any of the 15 types. Compute the expected # of different colors that are obtained in a set of 5 crayons. (Hint: use indicator variables and linearity of expectation)

We enumerate the crayons from 1 to 15. Let \(X_i\) indicate when the ith crayon is among the 5 crayons selected.

So,
\(E(X_i) =\) Pr {Probability that at least one type i crayon is in set of 5}
\(E(X_i) =\) 1 - Pr {no type i crayons in set of 5}
\(E(X_i) = 1 - \frac{14}{15}^5\ \)

Therefore, the expected # of crayons is:

\( = \sum_{i=1}^{25} E(X_i)\)
\( = 15[1 - \frac{14}{15}^5]\)
\( = 4.38\)

Sample question 2: Coding/computation

Given a dataframe, df, return only those rows which have missing values.
For example:

Will return...

Name	age	favorite_color	grade	name
Willard Morris	20	blue		Willard Morris
	22	yellow	95	Omar Mullins

 #Written in Python (Pandas)

 #First, we build a boolean series of the null values, using 'isnull' and 'any'
 #-->df.isnull().any(axis=1) will return the series True, False, True, False
 #We can then index this series against our dataframe to filter on the null values
 df[df.isnull().any(axis=1)]

Sample question 3: Coding/computation

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Given a single number, n, write a function using Python to return whether or not the number is prime. Additionally, if the inputted number is prime, save it into an array, a.

We'll set up a function below to determine whether or not a given number is prime, using simple if/else statements. Additionally, when a number is defined as prime we'll append it to our array, a.

#First, define an empty array to store prime numbers
a = []
#Define a function to identify whether or not a given number, x, is prime
def is_prime(x):
    if x < 2:
        #if the number is < 2, it's not prime, per definition of prime number 
        #(e.g. natural number greater than 1)
        return False
    else:
        #for all other numbers >=2
        for n in range(2,x):
            #if divisible by two smaller #s, then not prime
            if x % n == 0:
               return False
        #s that don't meet the above conditions are prime! save them to our array, a
        a.append(x)
        return True

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