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Unit 02 /
Problem 2:


I independent random variables
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Hard


(Concert Time)

You are excited about a concert featuring your favorite a cappella group: the Pitch Estimators. Tickets go on sale at noon, but before you can buy a ticket, you have to wait for your turn in an online waiting room. Because tickets are in high demand, you enlist two of your friends to help you. All three of you enter the waiting room at noon, and as soon as any one of you gets a ticket, you are done and can all sign off.

Suppose your waiting time in minutes is a continuous random variable $T$. Your first friend's waiting time in minutes is a continuous random variable $U$. Your second friend's waiting time in minutes is a continuous random variable $V$.

Suppose these random variables are are mutually independent and have probability density functions given by:

$$f_T(t)=\begin{cases} \frac{1}{3}e^{-\frac{1}{3}t}, &t \geq 0\ 0, &otherwise \end{cases}$$

$$f_U(t)=\begin{cases} \frac{1}{2}e^{-\frac{1}{2}t}, &t \geq0\ 0, &otherwise \end{cases}$$

$$f_V(t)=\begin{cases} \frac{1}{6}e^{-\frac{1}{6}t}, &t\geq 0\ 0, &otherwise \end{cases}$$

These are examples of what we call exponential random variables.

a. Please sketch the probability density for all three random variables on one graph.

b. For a particular time $t$, compute the probability that $T>t$, $U>t$, and $V>t$.

c. Let $O$ represent your overall waiting time. $O$ is the minimum of the waiting times for each of the three websites, $O = min(T,U,V)$. Compute the probability density function of $O$. (Hint: use the previous answer to write down the cdf)

Solution:

b. By independence, we have,

$$P(\{T>t\} \cap \{U>t\} \cap \{V>t\} ) = P(\{T>t\}) P( \{U>t\} ) P( \{V>t\} )$$

For the first multiplicand,

$$ P( \{T>t\} ) = \int_t^\infty f_T(u) du = \int_t^\infty \frac{1}{3}e^{-\frac{1}{3}u} = -e^{-\frac{1}{3}u} |_t^\infty = e^{-\frac{1}{3}t} $$

By similar argument,

$$ P( \{U>t\} ) = e^{-\frac{1}{2}t}, \qquad P( \{V>t\} ) = e^{-\frac{1}{6}t}$$

Therefore,

$$P(\{T>t\} \cap \{U>t\} \cap \{V>t\} ) = e^{-\frac{1}{3}t} e^{-\frac{1}{2}t} e^{-\frac{1}{6}t} = e^{-t} $$

c. Let $F_O$ be the cdf of $O$ and let $f_O$ be the pdf. For $t \ge 0$, we can write,

$$F_O(t) = P(O \le t) = P( min(T,U,V) \le t) = 1 - P( min(T,U,V) > t)$$

Where the last equality follows from the complement rule.

Finally, we note that the minimum of three numbers is bigger than $t$ whenever all three numbers are bigger than $t$, $\{ min(T,U,V) > t\} = \{T>t\} \cap \{U>t\} \cap \{V>t\}$. We can write,

$$F_O(t) = 1 - P( \{T>t\} \cap \{U>t\} \cap \{V>t\} ) = 1 - e^{-t} $$

To find the density, we take the derivative,

$$f(t) = \frac{d}{dt}F(t) = \frac{d}{dt} 1 - e^{-t} = e^{-t}$$