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 \(F(x)\). Suppose that Martian dice are 4-sided (tetrahedra) with points labeled 1 to 4. When a pair of these dice is tossed, let \(x\) be the product of the two numbers at the tops of the dice if the product is odd; otherwise \(x=0\).

Short answer

List all rolled outcomes and their probabilities. Compute mean and variance, and plot the CDF.

Step-by-step solution

- Sample Space and Random Variable

List all the possible outcomes of rolling two 4-sided dice. For each outcome, calculate the value of the random variable \( x \). If the product of the two numbers is odd, \( x \) is the product; otherwise, \( x = 0 \). Also, calculate the probability for each outcome which is \( \frac{1}{16} \).

- Create the Table

Construct a table listing each sample point, the corresponding value of \( x \), and its probability. Summarize this information into a table form for easier understanding.

- Values and Probabilities of \( x \)

List the different values of \( x \) and their corresponding probabilities \( p_{i} = f(x_{i}) \). Summarize this information into a frequency table.

- Compute the Mean (Expected Value)

The mean (expected value) \( \mu \) of \( x \) can be computed using the formula \( \mu = E(x) = \sum_{i} x_{i} p(x_{i}) \).

- Compute the Variance

The variance \( \sigma^2 \) of \( x \) is computed using the formula \( \sigma^2 = \sum_{i} (x_{i} - \mu)^2 p(x_{i}) \).

- Compute the Standard Deviation

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

- Find and Plot the Cumulative Distribution Function \( F(x) \)

The cumulative distribution function (CDF) \(F(x)\) is found by summing the probabilities \(P(X\leq x)\) for all values less than or equal to \( x \). Plot the CDF.

Key concepts

random variable

A random variable is a numerical outcome of a random phenomenon. Here, the random variable \( x \) represents the product of the numbers on the tops of two 4-sided dice, but only if the product is odd. If the product is even, \( x \) is assigned a value of 0.
This assignment of outcomes to numbers makes it easier to work with probabilistic situations mathematically. Random variables can take on different values with certain probabilities.
They help translate real-world scenarios into mathematical language, allowing for predictions and analysis of various events. For instance, rolling two Martian dice results in random values for \( x \) based on the outcomes.

expected value

The expected value (or mean) of a random variable, denoted as \( E(x) \) or \( \mu \), is essentially the long-term average value of the variable over many trials of the random process. For example, if we repeatedly roll two 4-sided dice and calculate values for \( x \), the average of all these values would converge to the expected value.
Mathematically, it is given by the formula:
\[ \mu = E(x) = \sum_{i} x_{i} p(x_{i}) \]
where \( x_{i} \) are possible values of \( x \) and \( p(x_{i}) \) are their corresponding probabilities. This gives us a weighted average, summing up all possible outcomes.

variance

Variance, denoted by \( \sigma^{2} \), measures how much the values of a random variable differ from the expected value.
It provides insight into the variability or dispersion of the data points. Mathematically:
\[ \sigma^{2} = \sum_{i}(x_{i} - \mu )^{2}p(x_{i}) \]
This formula takes each possible value of \( x \), subtracts the mean \( \mu \), squares the result, and multiplies by the probability of that outcome. Summing these gives the variance.
A high variance indicates data points are spread out over a wide range, whereas a low variance indicates they are closer to the mean.

standard deviation

The standard deviation, denoted by \( \sigma \), is the square root of the variance.
It expresses the average distance of each data point from the mean in the same units as the data points themselves. Mathematically:
\[ \sigma = \sqrt{\sigma^{2}} \]
This measure is more interpretable than the variance, as it provides a sense of typical distance from the mean.
For example, a small standard deviation means that most values are close to the mean, while a larger one indicates a wider spread around the mean.
Standard deviation is a crucial concept in understanding data distribution and variability.

cumulative distribution function

The cumulative distribution function (CDF) \( F(x) \) gives the probability that the random variable \( X \) is less than or equal to \( x \).
It is found by summing the probabilities of all outcomes up to and including \( x \). Mathematically:
\[ F(x) = P(X \leq x) \]
This function helps understand the probability distribution of a random variable more intuitively.
By plotting \( F(x) \), we can visualize how probabilities accumulate as the value of \( x \) increases. For example, if we plot the CDF of the random variable \( x \) from our Martian dice scenario, we would see a step function representing the cumulative probabilities at each value of \( x \).