Question

For the random variables described, find and graph the probability distribution for \(x .\) Then calculate the mean, variance, and standard deviation. Toss a pair of dice and record \(x,\) the sum of the numbers on the two upper faces.

Short answer

Answer: The probability distribution is as follows: - \(P(x=2) = \frac{1}{36}\) - \(P(x=3) = \frac{2}{36}\) - \(P(x=4) = \frac{3}{36}\) - \(P(x=5) = \frac{4}{36}\) - \(P(x=6) = \frac{5}{36}\) - \(P(x=7) = \frac{6}{36}\) - \(P(x=8) = \frac{5}{36}\) - \(P(x=9) = \frac{4}{36}\) - \(P(x=10) = \frac{3}{36}\) - \(P(x=11) = \frac{2}{36}\) - \(P(x=12) = \frac{1}{36}\) The mean (\(\mu\)) is 7, the variance (\(\sigma^2\)) is \(\frac{105}{6}\), and the standard deviation (\(\sigma\)) is \(\frac{\sqrt{105}}{\sqrt{6}}\).

Step-by-step solution

Find all possible outcomes

To find all the possible outcomes of tossing a pair of dice, we can list all the possible combinations of the numbers on the two dice. Since each die has six numbered faces (1 to 6), there are 36 possible outcomes.

Determine the probability of each outcome

Next, we will find the probability of each sum from 2 to 12. We'll count how many times each sum appears in the 36 possible outcomes, and then divide it by 36 to get the probability. Probabilities: - \(P(x=2) = \frac{1}{36}\) - \(P(x=3) = \frac{2}{36}\) - \(P(x=4) = \frac{3}{36}\) - \(P(x=5) = \frac{4}{36}\) - \(P(x=6) = \frac{5}{36}\) - \(P(x=7) = \frac{6}{36}\) - \(P(x=8) = \frac{5}{36}\) - \(P(x=9) = \frac{4}{36}\) - \(P(x=10) = \frac{3}{36}\) - \(P(x=11) = \frac{2}{36}\) - \(P(x=12) = \frac{1}{36}\)

Calculate the mean, variance, and standard deviation

To calculate the mean, we'll multiply each outcome by its probability and add them up: \(E(x)=\mu =\sum_{i=2}^{12}x_iP(x_i) = \frac{1(2)+2(3)+3(4)+4(5)+5(6)+6(7)+5(8)+4(9)+3(10)+2(11)+1(12)}{36}=7\) To find the variance and standard deviation, we first need to compute \(E(x^2)\): \(E(x^2)=\sum_{i=2}^{12}x_i^2P(x_i)=\frac{1(2^2)+2(3^2)+3(4^2)+4(5^2)+5(6^2)+6(7^2)+5(8^2)+4(9^2)+3(10^2)+2(11^2)+1(12^2)}{36}=\frac{490}{6}\) Next, we compute the variance \(\sigma^2\): \(\sigma^2 = E(x^2)-(\mu)^2 = \frac{490}{6} - (7)^2 = \frac{105}{6}\) Finally, we find the standard deviation \(\sigma\) by taking the square root of the variance: \(\sigma = \sqrt{\sigma^2} = \sqrt{\frac{105}{6}} = \frac{\sqrt{105}}{\sqrt{6}}\)

Graph the probability distribution

To graph the probability distribution, we will plot each sum (from 2 to 12) on the x-axis and its corresponding probability on the y-axis. We will see that the graph forms a symmetric, bell-shaped curve centered at the mean value of 7.

Key concepts

Probability Theory

Probability theory is the mathematical framework for analyzing experiments and processes that produce random outcomes. When we toss a pair of dice, for example, the result is uncertain because it can be any number from 2 to 12. Each possible sum of the dice faces represents an event with a certain probability—that is, the likelihood of its occurrence.

The outcomes in the exercise at hand are possible sums we get from rolling two dice. By listing all combinations, we found that there are 36 different outcomes. We then calculated the chance of rolling a particular sum (for instance, a 7) by finding how many combinations result in that sum. The fundamental aspect to grasp here is that each outcome's probability can be described mathematically, and when considered together, they form what we call a 'probability distribution' for the random variable, in this case, the sum of the two dice.

Mean Calculation

The mean, often referred to as the expected value in probability, is an average that takes into account the likelihood of various outcomes. This contrasts with a simple average that treats all outcomes the same.

In this dice example, to calculate the mean, we multiply each potential sum by its chance of occurring and add these products together. Because the probabilities are not uniform—certain sums occur more frequently than others—the mean gives us a 'weighted average.' This average, or expected value, provides a central or typical value for the probability distribution and, in our case, it is 7. It's an essential concept, as the mean serves as a balance point for the probability distribution of the random variable.

Variance and Standard Deviation

Variance and standard deviation are two critical measures of how spread out the values in a probability distribution are. The variance is the average squared deviation of each possible outcome from the mean, which provides us with a measure of how broadly the random variable's values are dispersed.

The steps in the problem guide you through the calculation of variance, which involves squaring each outcome, weighting it by its probability, and then, summing these numbers—a process akin to calculating the mean, but for the squared outcomes. The standard deviation, then, is simply the square root of the variance. It represents the average distance from the mean, making it a more immediately understandable measure of spread because it is in the same units as the random variable itself—in our dice example, the number of dots showing on the dice.

Graphing Probability Distributions

Graphing a probability distribution visually represents the probabilities of different outcomes. For the sum of two dice, we create a discrete probability distribution graph since the sums are whole numbers and not continuous. Each sum is represented on the x-axis, while the probability of each sum is on the y-axis.

By plotting the points and connecting them, we might notice a symmetrical, bell-shaped curve for the dice sum distribution, peaking at the mean of 7. This graph helps in visualizing the concept of probability distribution, showing that certain outcomes are more likely than others, and illustrates how the probability decreases symmetrically as we move away from the mean. It is not only an effective tool for communication but also aids in understanding and interpreting the significance of the measures of central tendency and dispersion in the context of the given data.