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:

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

Sample question 2: Coding/computation

Suppose you have a dataframe, df, with the following records:

The dataframe is showing information about students. Write code using Python Pandas to select the rows where the students' favorite color is blue or yellow and their grade is at least 90.

Click here to view this problem in an interactive Colab (Jupyter) notebook.

#define array of target colors
fav_color_filter = ['blue', 'yellow']
#To select rows whose column value is in an iterable array, which we defined as fav_color_filter, we can use isin
df = df.loc[df['favorite_color'].isin(fav_color_filter)]
#next, we need to filter on scores above 90. here we can use loc on our dataframe:
df = df.loc[(df['grade'] >= 90)]
#preview the dataframe
df.head()

Resultant dataframe:

Click here to view this solution in an interactive Colab (Jupyter) notebook.

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.

Click here to view this problem in an interactive Colab (Jupyter) notebook.

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

Click here to view this solution in an interactive Colab (Jupyter) notebook.

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