1. Discrete Random Variables. Suppose that we are interested in the number of cups of coffee drank by a (randomly selected) student at UCLA. This quantity can be represented as a random variable Y with probability mass function:

if a ∈ {0,1,2}

if a = 3

if a = 4 ,

if a = 5 0 otherwise

where c is an unknown constant.

(a) Explain why the number of cups of coffee drank in a day by a randomly selected student at UCLA is a random variable.

(b) What is the relevant outcome space of the random variable Y ?

(c) Explain what the distribution of this random variable represents. In other words distribution of Y assigns a probability to any subset of the outcome space. How do we interpret this probability? (d) Solve for c. (Hint: Recall that PY (OY ) = 1 so that Pa∈OY pY (a) must equal one).

(e) What is the probability that a randomly selected student at UCLA drinks at least 3 cups of coffee a day, PY (Y ≥ 3)?

(f) What is the expected number of cups of coffee drank per day for a randomly selected student at UCLA?

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