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Chapter 15: Problem 1

Set up sample spaces for Problems 1 to 7 and list next to each sample point the value of the indicated random variable \(x,\) and the probability associated with the sample point. Make a table of the different values \(x_{i}\) of \(x\) and the corresponding probabilities \(p_{i}=f\left(x_{i}\right)\) Compute the mean, the variance, and the standard deviation for \(x\). Find and plot the cumulative distribution function \(F(x)\). Three coins are tossed; \(x=\) number of heads minus number of tails.

Short Answer

Expert verified
Mean = 0, Variance = 2, Standard Deviation ≈ 1.41.

Step by step solution

01

Identify Sample Space

When three coins are tossed, the possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. Write down all eight possible outcomes.
02

Define the Random Variable

Let the random variable x be the number of heads minus the number of tails in each outcome. Calculate x for each of the sample points: HHH (3), HHT (1), HTH (1), HTT (-1), THH (1), THT (-1), TTH (-1), TTT (-3).
03

Assign Probabilities

Each of the eight outcomes has an equal probability of occurring. Thus, each sample point has the probability 1/8. Create a table listing each outcome, the value of x, and its probability.
04

Create the Probability Distribution Table

List the unique values of x and determine their corresponding probabilities. For x = -3, p = 1/8; for x = -1, p = 3/8; for x = 1, p = 3/8; for x = 3, p = 1/8. Create a table showing x_i and p_i=f(x_i).
05

Compute the Mean (\mu)

The mean or expected value of x is calculated as \(\mu = \sum_{i} x_{i} p_{i}\). Using the values, \mu = (-3 \times 1/8) + (-1 \times 3/8) + (1 \times 3/8) + (3 \times 1/8) = 0.
06

Compute the Variance (\sigma^2)

The variance is calculated as \(\sigma^{2} = \sum_{i} (x_{i} - \mu)^{2} p_{i}\). Using the mean value, \sigma^{2} = ((-3 - 0)^{2} \times 1/8) + ((-1 - 0)^{2} \times 3/8) + ((1 - 0)^{2} \times 3/8) + ((3 - 0)^{2} \times 1/8) = 2.
07

Compute the Standard Deviation (\sigma)

The standard deviation is the square root of the variance, \(\sigma = \sqrt{\sigma^{2}} = \sqrt{2} \approx 1.41\).
08

Plot the Cumulative Distribution Function (F(x))

The cumulative distribution function F(x) is the sum of the probabilities for all values less than or equal to a certain x. Create the table and plot the following points: F(-3) = 1/8, F(-2) = 1/8, F(-1) = 4/8, F(0) = 4/8, F(1) = 7/8, F(2) = 7/8, F(3) = 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Space
In probability, the sample space is a comprehensive list of all possible outcomes of a random experiment. When three coins are tossed, the sample space includes all potential combinations of heads (H) and tails (T). Therefore, the sample space can be represented as follows:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. There are eight different outcomes. Each outcome is equally likely, which forms the foundation for calculating probabilities in subsequent steps.
Random Variable
A random variable is a numerical representation of the outcomes of a random experiment. In our exercise, we define the random variable x as the number of heads minus the number of tails. This gives us a way to assign numerical values to each outcome in the sample space.
For instance, in the outcome HHH, we have 3 heads and 0 tails, giving us a random variable value of 3 (i.e., 3-0 = 3). Similarly, for the outcome HTT, we have 1 head and 2 tails, resulting in a random variable value of -1 (i.e., 1-2 = -1).
Mean

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Most popular questions from this chapter

(a) A candy vending machine is out of order. The probability that you get a candy bar (with or without return of your money) is \(\frac{1}{2},\) the probability that you get your money back (with or without candy) is \(\frac{1}{3}\), and the probability that you get both the candy and your money back is \(\frac{1}{12}\). What is the probability that you get nothing at all? Suggestion: Sketch a geometric diagram similar to Figure 3.1, indicate regions representing the various possibilities and their probabilities; then set up a four-point sample space and the associated probabilities of the points. (b) Suppose you try again to get a candy bar as in part (a). Set up the 16 -point sample space corresponding to the possible results of your two attempts to buy a candy bar, and find the probability that you get two candy bars (and no money back); that you get no candy and lose your money both times; that you just get your money back both times.

Show that adding a constant \(K\) to a random variable increases the average by \(K\) but does not change the variance. Show that multiplying a random variable by \(K\) multiplies both the average and the standard deviation by \(K\).

The following game was being played on a busy street: Observe the last two digits on each license plate. What is the probability of observing at least two cars with the same last two digits among the first 5 cars? 10 cars? 15 cars? How many cars must you observe in order for the probability to be greater than \(\frac{1}{2}\) of observing two with the same last two digits?

There are 3 red and 2 white balls in one box and 4 red and 5 white in the second box. You select a box at random and from it pick a ball at random. If the ball is red, what is the probability that it came from the second box?

Let \(\mu\) be the average of the random variable \(x\). Then the quantities \(\left(x_{i}-\mu\right)\) are the deviations of \(x\) from its average. Show that the average of these deviations is zero. Hint: Remember that the sum of all the \(p_{i}\) must equal 1.
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