### Probability Distribution

**Probability Distribution for a Discrete** Random **Variable**

The example below illustrates how to specify the possible values that a discrete random variable can assume.

#### Constructing a Probability Distribution

Suppose you simultaneously toss two fair coins. Let \begin{align*}X\end{align*} be the number of heads observed. Find the probability associated with each value of the random variable \begin{align*}X\end{align*}.

Since there are two coins, and each coin can be either heads or tails, there are four possible outcomes \begin{align*}(HH, HT, TH, TT)\end{align*}, each with a probability of \begin{align*}\frac{1}{4}\end{align*}. Since \begin{align*}X\end{align*} is the number of heads observed, \begin{align*}x= 0, 1, 2.\end{align*}

We can identify the probabilities of the simple events associated with each value of \begin{align*}X\end{align*} as follows:

\begin{align*}P(x=0) &= P(TT)=\frac{1}{4}\\ P(x=1) &= P(HT)+P(TH)=\frac{1}{4}+\frac{1}{4}=\frac{1}{2}\\ P(x=2) &= P(HH)=\frac{1}{4}\end{align*}

This is a complete description of all the possible values of the random variable, along with their associated probabilities. We refer to this as a *probability distribution.* This probability distribution can be represented in different ways. Sometimes it is represented in tabular form and sometimes in graphical form. Both forms are shown below.

In tabular form:

\begin{align*}x\end{align*} | \begin{align*}P(x)\end{align*} |
---|---|

0 | \begin{align*}\frac{1}{4}\end{align*} |

1 | \begin{align*}\frac{1}{2}\end{align*} |

2 | \begin{align*}\frac{1}{4}\end{align*} |

**Figure:** The tabular form of the probability distribution for the random variable in the first example.

As a graph:

A probability distribution of a random variable specifies the values the random variable can assume, along with the probability of it assuming each of these values. All probability distributions must satisfy the following two conditions:

\begin{align*}& P(x) \ge 0, \text{for all values of} \ X\\ & \sum_{} P(x)=1, \text{for all values of} \ X\end{align*}

#### Determining a Probability Distribution

What is the probability distribution for the number of yes votes for three voters? (See the first example in the Chapter Introduction.)

Since each of the 8 outcomes is equally likely, the following table gives the probability of each value of the random variable. The value of the random variable is the *number of yes votes*.

Value of Random Variable |
Probability |
---|---|

3 | \begin{align*}\frac{1}{8} = 0.125\end{align*} |

2 | \begin{align*}\frac{3}{8}=0.375\end{align*} |

1 | \begin{align*}\frac{3}{8}=0.375\end{align*} |

0 | \begin{align*}\frac{1}{8} = 0.125\end{align*} |

**Figure:** Tabular representation of the probability distribution for the random variable in the first example in the Chapter Introduction.

#### Determining the Relationship Between Observed Values and Random Variables

Consider the following two probability distributions:

X | 2 | 3 | 4 |

P(X) | 1/5 | 1/5 | 3/5 |

Y | 3 | 4 | 5 |

P(Y) | 3/5 | 1/5 | 1/5 |

The observed values are 3 and 4. Which observed value goes with which random variable? Are you sure? Explain.

The observed value 3 is more likely to come from the second probability distribution. In this distribution the probability of obtaining a 3 is 3/5. In the first distribution the probability of obtaining a 3 is 1/5. The value 4 has a probability of 3/5 in the first distribution but only a probability of 1/5 in the second distribution. However, you can not be certain which distribution these values comes from.

### Example

#### Example 1

Consider the following two probability distributions:

X | 1 | 2 | 3 |

P(X) | 1/3 | 1/3 | 1/3 |

Y | 4 | 5 | |

P(Y) | 1/3 | 2/3 |

There are two observed values: 3 and 4. Which observed value goes with which random variable? Are you sure? Explain.

The 3 is a possible value of the random variable X. The random variable Y does not take on the value of 3. The value 4 is a possible value of the random variable Y. The random variable X does not take on the value 4.

### Review

- Consider the following probability distribution: \begin{align*}& x && -4 && 0 && 1 && 3\\
& P(x) && 0.1 && 0.3 && 0.4 && 0.2\end{align*}
- What are all the possible values of \begin{align*}X\end{align*}?
- What value of \begin{align*}X\end{align*} is most likely to happen?
- What is the probability that \begin{align*}x > 0\end{align*}?
- What is the probability that \begin{align*}x = -2\end{align*}?

- A fair die is tossed twice, and the up face is recorded each time. Let \begin{align*}X\end{align*}be the sum of the up faces.
- Give the probability distribution for \begin{align*}X\end{align*} in tabular form.
- What is \begin{align*}P(x \ge 8)\end{align*}?
- What is \begin{align*}P(x < 8)\end{align*}?
- What is the probability that \begin{align*}x\end{align*} is odd? What is the probability that \begin{align*}x\end{align*} is even?
- What is \begin{align*}P(x=7)\end{align*}?

- If a couple has three children, what is the probability that they have at least one boy?

- Suppose there are six numbers in a box: 1, 2, 3, 4, 5, 6.
- Suppose you draw two numbers with replacement. Are the draws independent? Explain.
- Suppose you draw two numbers without replacement. Are the draws independent? Explain.

- Two draws are made at random without replacement from a box with four numbers: 1,2, 3, 4. Find the probability that the second draw will be a 3. Explain.
- Suppose there is a box with four slips of paper each paper with one number: 1, 2, 3, 3. Let the random variable \begin{align*}X\end{align*} be defined as the number you choose at random. What is \begin{align*}P(X=1)\end{align*} ?
- Suppose a box has four slips of paper and on each slip are two numbers. The slips of paper look like the following: \begin{align*}
& X && Y\\
& 1 && 3\\
& 2 && 2\\
& 3 && 1\\
& 5 && 1
\end{align*}
- Explain in words, what \begin{align*}X\cdot Y\end{align*} means.
- Find \begin{align*}P(X\cdot Y=3)\end{align*}.
- Find \begin{align*}P(2X-3Y=7)\end{align*}.

- True or False? If \begin{align*}X\end{align*} and \begin{align*}Y\end{align*} are independent then \begin{align*}Y\end{align*} and \begin{align*}X\end{align*} are independent.
- Suppose two draws will be made at random with replacement from a box that has three slips of paper, each with a number on it: 1, 2, and 3. Let \begin{align*}X_1\end{align*} represent the first draw and \begin{align*}X_2\end{align*}represent the second draw.
- What is \begin{align*}P(X_1=1)\end{align*}?
- Find the chance that the first draw will be a one and the second draw will be a 2.
- Find \begin{align*}P(X_1=1)\cdot P(X_2=2)\end{align*}
- Are and independent? Explain.
- Now suppose two draws are made at random without replacement. Are the variables independent? Explain.

- Suppose the random variable \begin{align*}X\end{align*} can take on the values 1 and 2 and the random variable \begin{align*}Y\end{align*} can take on the values 1 and 3. If you are to be paid whatever value the random variable turns out to be, in dollars, which random variable do you prefer? Explain.
- Suppose \begin{align*}f(x)=\frac{a}{x^2+1}\end{align*} for \begin{align*}x=0,1,2,3\end{align*}is a discrete probability distribution.
- Find \begin{align*}a\end{align*}
- Find \begin{align*}P(x>0)\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.2.