6.2 Problems on Random Variables and Probabilities

The following simple random variable is in canonical form:

$X = - 3 . 75 I_{A} - 1 . 13 I_{B} + 0 I_{C} + 2 . 6 I_{D}$.

Express the events $\left\{X \in \left(- 4 , 2\right]\right\} , \left\{X \in \left(0 , 3\right]\right\} , \left\{X \in \left(- \infty , 1\right]\right\}$, $\left\{\left|X - 1\right| \geq 1\right\}$, and $\left\{X \geq 0\right\}$ in terms of $A , B , C$, and D.

Random variable X, in canonical form, is given by $X = - 2 I_{A} - I_{B} + I_{C} + 2 I_{D} + 5 I_{E}$.

Express the events $\left\{X \in \left[2 , 3\right)\right\} , \left\{X \leq 0\right\} , \left\{X < 0\right\}$, $\left\{\left|X - 2\right| \leq 3\right\}$, and $\left\{X^{2} \geq 4\right\}$, in terms of $A , B , C , D$, and E.

The class $\left\{C_{j} : 1 \leq j \leq 10\right\}$ is a partition. Random variable X has values $\left\{1 , 3 , 2 , 3 , 4 , 2 , 1 , 3 , 5 , 2\right\}$ on C₁ through C₁₀, respectively. Express X in canonical form.

The class $\left\{C_{j} : 1 \leq j \leq 10\right\}$ in the exercise has respective probabilities 0.08, 0.13, 0.06, 0.09, 0.14, 0.11, 0.12, 0.07, 0.11, 0.09. Determine the distribution for X.

Suppose $X , Y$ in canonical form are

The $P \left(A_{i}\right)$ are 0.3, 0.6, 0.1, respectively, and the $P \left(B_{j}\right)$ are 0.2 0.6 0.2. Each pair $\left\{A_{i} , B_{j}\right\}$ is independent. Consider the random variable $Z = X + Y$. Then $Z = 2 + 1$ on $A_{1} B_{1}$, $Z = 3 + 3$ on $A_{2} B_{3}$, etc. Determine the value of Z on each $A_{i} B_{j}$ and determine the corresponding $P \left(A_{i} B_{j}\right)$. From this, determine the distribution for Z.

For the random variables in the exercise, let $W = X Y$. Determine the value of W on each $A_{i} B_{j}$ and determine the distribution of W.

A pair of dice is rolled.

1. Let X be the minimum of the two numbers which turn up. Determine the distribution for X
2. Let Y be the maximum of the two numbers. Determine the distribution for Y.
3. Let Z be the sum of the two numbers. Determine the distribution for Z.
4. Let W be the absolute value of the difference. Determine its distribution.

Minterm probabilities $p \left(0\right)$ through $p \left(15\right)$ for the class $\left\{A , B , C , D\right\}$ are, in order,

Determine the distribution for random variable

On a Tuesday evening, the Houston Rockets, the Orlando Magic, and the Chicago Bulls all have games (but not with one another). Let A be the event the Rockets win, B be the event the Magic win, and C be the event the Bulls win. Suppose the class $\left\{A , B , C\right\}$ is independent, with respective probabilities 0.75, 0.70 0.8. Ellen's boyfriend is a rabid Rockets fan, who does not like the Magic. He wants to bet on t…

The class $\left\{A , B , C , D\right\}$ has minterm probabilities

• Determine whether or not the class is independent.
• The random variable $X = I_{A} + I_{B} + I_{C} + I_{D}$ counts the number of the events which occur on a trial. Find the distribution for X and determine the probability that two or more occur on a trial. Find the probability that one or three of these occur on a trial.

James is expecting three checks in the mail, for $20, $26, and $33 dollars. Their arrivals are the events $A , B , C$. Assume the class is independent, with respective probabilities 0.90, 0.75, 0.80. Then

represents the total amount received. Determine the distribution for X. What is the probability he receives at least $50? Less than $30?

A gambler places three bets. He puts down two dollars for each bet. He picks up three dollars (his original bet plus one dollar) if he wins the first bet, four dollars if he wins the second bet, and six dollars if he wins the third. His net winning can be represented by the random variable

Assume the results of the games are independent. Determine the distribution for X.

Henry goes to a hardware store. He considers a power drill at $35, a socket wrench set at $56, a set of screwdrivers at $18, a vise at $24, and hammer at $8. He decides independently on the purchases of the individual items, with respective probabilities 0.5, 0.6, 0.7, 0.4, 0.9. Let X be the amount of his total purchases. Determine the distribution for X.

A sequence of trials (not necessarily independent) is performed. Let E_i be the event of success on the ith component trial. We associate with each trial a “payoff function” $X_{i} = a I_{E_{i}} + b I_{E_{i}^{c}}$. Thus, an amount a is earned if there is a success on the trial and an amount b (usually negative) if there is a failure. Let S_n be the number of successes in the n trials and W be the net payoff. Show that $W = \left(a - b\right) S_{n} + b n$.

A marker is placed at a reference position on a line (taken to be the origin); a coin is tossed repeatedly. If a head turns up, the marker is moved one unit to the right; if a tail turns up, the marker is moved one unit to the left.

1. Show that the position at the end of ten tosses is given by the random variable
   $X = \sum_{i = 1}^{10} I_{E_{i}} - \sum_{i = 1}^{10} I_{E_{i}^{c}} = 2 \sum_{i = 1}^{10} I_{E_{i}} - 10 = 2 S_{10} - 10$
   where E_i is the event of a head on the ith toss and S₁₀ is the number of heads in ten…