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Probability Distribution

Probabilities associated with possible counting number

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Probability Distribution

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

  1. Consider the following probability distribution: \begin{align*}& x && -4 && 0 && 1 && 3\\ & P(x) && 0.1 && 0.3 && 0.4 && 0.2\end{align*}
    1. What are all the possible values of \begin{align*}X\end{align*}?
    2. What value of \begin{align*}X\end{align*} is most likely to happen?
    3. What is the probability that \begin{align*}x > 0\end{align*}?
    4. What is the probability that \begin{align*}x = -2\end{align*}?
  2. 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.
    1. Give the probability distribution for \begin{align*}X\end{align*} in tabular form.
    2. What is \begin{align*}P(x \ge 8)\end{align*}?
    3. What is \begin{align*}P(x < 8)\end{align*}?
    4. What is the probability that \begin{align*}x\end{align*} is odd? What is the probability that \begin{align*}x\end{align*} is even?
    5. What is \begin{align*}P(x=7)\end{align*}?
  3. If a couple has three children, what is the probability that they have at least one boy?
  1. Suppose there are six numbers in a box: 1, 2, 3, 4, 5, 6.
    1. Suppose you draw two numbers with replacement. Are the draws independent? Explain.
    2. Suppose you draw two numbers without replacement. Are the draws independent? Explain.
  2. 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.
  3. 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*} ?
  4. 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*}
    1. Explain in words, what \begin{align*}X\cdot Y\end{align*} means.
    2. Find \begin{align*}P(X\cdot Y=3)\end{align*}.
    3. Find \begin{align*}P(2X-3Y=7)\end{align*}.
  5. 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.
  6. 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.
    1. What is \begin{align*}P(X_1=1)\end{align*}?
    2. Find the chance that the first draw will be a one and the second draw will be a 2.
    3. Find \begin{align*}P(X_1=1)\cdot P(X_2=2)\end{align*}
    4. Are and independent? Explain.
    5. Now suppose two draws are made at random without replacement. Are the variables independent? Explain.
  7. 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.
  8. 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.
    1. Find \begin{align*}a\end{align*}
    2. 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. 

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Vocabulary

Histogram

A histogram is a display that indicates the frequency of specified ranges of continuous data values on a graph in the form of immediately adjacent bars.

probability distribution

In a probability distribution, you may have a table, a graph, or a chart that shows you all the possible values of X (your variable), and the probability associated with each of these values P(X).

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