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Question 5: There is a song-writing contest at your local radio station. Everyone who enters this contest writes their song using Sing-GPT, an AI song writing service, to generate their songs. Of course, since Sing-GPT is not terribly original, it only generates 100 different songs. There are 50 people entering into the contest, each with a uniformly random generated song chosen from the 100 different songs available. Hint: For both questions below, use indicator random variables. a) Let X be the number of different songs entered into the contest. What is E(X)? b) If a song is entered more than once, it is ejected from the contest. What is the expected number of songs to be ejected? 3 The next two questions use the following definition. If X is a random variable that can take any value $n$ where $n$ is an integer and $n \ge 1$, and if A is an event, then the conditional expected value $E(X|A)$ is defined as $\infty$ $E(X|A) = \sum_{k=1} k \cdot Pr(X = k|A).$

          Question 5:
There is a song-writing contest at your local radio station. Everyone who enters this
contest writes their song using Sing-GPT, an AI song writing service, to generate their songs.
Of course, since Sing-GPT is not terribly original, it only generates 100 different songs. There
are 50 people entering into the contest, each with a uniformly random generated song chosen
from the 100 different songs available.
Hint: For both questions below, use indicator random variables.
a) Let X be the number of different songs entered into the contest. What is E(X)?
b) If a song is entered more than once, it is ejected from the contest. What is the expected
number of songs to be ejected?
3
The next two questions use the following definition. If X is a random variable that can
take any value $n$ where $n$ is an integer and $n \ge 1$, and if A is an event, then the conditional
expected value $E(X|A)$ is defined as
$\infty$
$E(X|A) = \sum_{k=1} k \cdot Pr(X = k|A).$

Question 5:
There is a song-writing contest at your local radio station. Everyone who enters this
contest writes their song using Sing-GPT, an AI song writing service, to generate their songs.
Of course, since Sing-GPT is not terribly original, it only generates 100 different songs. There
are 50 people entering into the contest, each with a uniformly random generated song chosen
from the 100 different songs available.
Hint: For both questions below, use indicator random variables.
a) Let X be the number of different songs entered into the contest. What is E(X)?
b) If a song is entered more than once, it is ejected from the contest. What is the expected
number of songs to be ejected?
3
The next two questions use the following definition. If X is a random variable that can
take any value n where n is an integer and n ≥ 1, and if A is an event, then the conditional
expected value E(X|A) is defined as
∞
E(X|A) = ∑k=1 k · Pr(X = k|A).

Added by Hector G.

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