Question

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

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

Step-by-step solution

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.

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).

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.

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).

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.

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.

Compute the Standard Deviation (\sigma)

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

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.

Key concepts

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