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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

Recall that two events \(A\) and \(B\) are called independent if \(p(A B)=p(A) p(B) .\) Similarly two random variables \(x\) and \(y\) are called independent if the joint probability function \(f(x, y)=g(x) h(y) .\) Show that if \(x\) and \(y\) are independent, then the expectation or average of \(x y\) is \(E(x y)=E(x) E(y)=\mu_{x} \mu_{y}\).

Some transistors of two different kinds (call them \(N\) and \(P\) ) are stored in two boxes. You know that there are \(6 N\) 's in one box and that \(2 N\) 's and \(3 P\) 's got mixed in the other box, but you don't know which box is which. You select a box and a transistor from it at random and find that it is an \(N ;\) what is the probability that it came from the box with the \(6 \mathrm{N}\) 's? From the other box? If another transistor is picked from the same box as the first, what is the probability that it is also an \(N ?\)

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 \(K\).

A circular garden bed of radius \(1 \mathrm{m}\) is to be planted so that \(N\) seeds are uniformly distributed over the circular area. Then we can talk about the number \(n\) of seeds in some particular area \(A,\) or we can call \(n / N\) the probability for any one particular seed to be in the area \(A\). Find the probability \(F(r)\) that a seed (that is, some particular seed) is within \(r\) of the center. (Hint: What is \(\mathrm{F}(1) ?\) ) Find \(f(r) d r,\) the probability for a seed to be between \(r\) and \(r+d r\) from the center. Find \(\bar{r}\) and \(\sigma\).

In a club with 500 members, what is the probability that exactly two people have birthdays on July 4?
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