Problems

01

Digit of a real number

Suppose a real number is chosen randomly in the unit interval $[\mathrm{0,1}]$. Consider the decimal expansion of this number. Let $Y$ be a random variable giving the first digit after the decimal point. Find the possible values, the PMF, and the CDF of $Y$.

02

Gambling with a coin

Two players, A and B, are flipping a fair coin together. If it comes up heads, A pays $1 to B, and if it comes up tails, B pays $1 to A.

They play five rounds. Let $X$ be a random variable recording A’s final winnings.

(a) Find the set of possible values of $X$. (I.e., the set of outcomes with nonzero probability.)

(b) Find the PMF and CDF of $X$.

03

PMF from CDF for tracking gasoline

I am trying to keep track of how much gasoline (measured in gallons, rounded to the nearest integer, and denoted by $X$) my car uses every week. I have managed to find the CDF of $X$:

$$F(x)=\{\begin{array}{cc}0& \phantom{0\le }x<0\\ 0.1& 0\le x<1\\ 0.2& 1\le x<2\\ 0.5& 2\le x<3\\ 1& x\ge 3\end{array}$$

Assume gasoline costs $3 a gallon. Let $Y$ denote the amount of money I spend on gasoline every week. What is the PMF of $Y$?

04

Patients in the hospital

After being discharged from the hospital following a particular surgery, patients often make visits to their local emergency room for treatment. The function below is the CDF of X, the number of emergency room visits per patient:

$$F_X(x)=\{\begin{array}{cc}0& \phantom{0\le }x<0\\ 0.57& 0\le x<1\\ 0.82& 1\le x<2\\ 0.97& 2\le x<3\\ 1& 3\le x\end{array}$$

(a) Find the probability a patient will make more than 1 visit to the emergency room.

(b) Find the probability a patient will not visit the emergency room.

05

Construct random variable from PMF

The probability space $S={\{H,T\}}^{\mathbb{N}}$ is the set of infinite strings of $H$ or $T$, for example $HTTHTHHTTH\cdots \in S$. There is a probability measure $P$ determined by the interpretation of $H$ or $T$ as the outcome of a fair coin flip. For a subset $A$ containing all strings with specified (given) values of the first $n$ letters, then $P(A)=2^{-n}$. E.g. if $A=\{HTTT\cdots ,HTTH\cdots ,HTHT\cdots ,\dots \}$ is everything starting with $HT$, then $P(A)=1/4$.

(a) Construct a random variable $X$ on $S$ such that the PMF of $X$ is given by:

$$P_X(k)=\{\begin{array}{cc}3/8& \text{for }k=0\\ 1/8& \text{for }k=5\\ 1/2& \text{for }k=17\\ 0& \text{else}\end{array}$$

(b) Repeat (a) but with a biased coin for which $P[H]=1/4$.

06

Variance from CDF: Drill bit changes

The bits for a particular kind of drill must be changed fairly often. Let $X$ denote the number of holes that can be drilled with one bit. The CDF of $X$ is given below:

$$F_X(x)=\{\begin{array}{cc}0& \phantom{0\le }x<1\\ 0.13& 1\le x<2\\ 0.48& 2\le x<3\\ 0.81& 3\le x<4\\ 1& 4\le x\end{array}$$

(a) Find the probability that a bit will be able to drill more than 2 holes.

(b) Find $\mathrm{Var}[X]$ by constructing the PMF.

07

Half the babies are female?

At Grace Community Hospital, 4 babies are delivered in one day. At Hope Valley Hospital, 6 babies are delivered in one day.

Consider these two events:

• (i) exactly half the babies born at Grace Community are female
• (ii) exactly half the babies born at Hope Valley are female

Perform a calculation to determine whether Event (i) is more probable, Event (ii) is more probable, or they are equally probable. (Assume the probability of each baby being born male or female is $0.5$.)