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 \(F(x)\). A weighted coin with probability \(p\) of coming down heads is tossed three times; \(x=\) number of heads minus number of tails.

Short answer

Define sample space, calculate \(x\), assign probabilities, create table, compute mean, variance, standard deviation, find and plot CDF.

Step-by-step solution

Define the sample space

List all possible outcomes when a weighted coin is tossed three times: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Each of these corresponds to an element in our sample space.

Calculate the value of the random variable

For each outcome, calculate the value of the random variable \(x = \text{number of heads} - \text{number of tails}\). Example calculations:- HHH: \(x = 3 - 0 = 3\)- HHT: \(x = 2 - 1 = 1\)- HTH: \(x = 2 - 1 = 1\)- HTT: \(x = 1 - 2 = -1\)- THH: \(x = 2 - 1 = 1\)- THT: \(x = 1 - 2 = -1\)- TTH: \(x = 1 - 2 = -1\)- TTT: \(x = 0 - 3 = -3\).

Assign probabilities to each outcome

Since the coin is weighted, the probability of heads is \(p\) and tails is \(1-p\). For each outcome, calculate the associated probability. For example, the probability for HHH is \(p^3\), for HHT is \(p^2(1-p)\), etc.

Create a probability table

Create a table showing the possible values of \(x_i\) and their corresponding probabilities \(p_i\). Sum the probabilities for repeated values of \(x_i\). Example table:| \(x_i\) | Value | Probability ||------|---------|--------------------|| 3 | 3 - 0 | \(p^3\) || 1 | 2 - 1 | \(3p^2(1-p)\) || -1 | 1 - 2 | \(3p(1-p)^2\) || -3 | 0 - 3 | \( (1-p)^3 \)

Compute the mean

The mean of \(x\) is given by \(\mu = \sum x_i p_i\). Using values from the table, calculate \(\mu\).

Compute the variance

The variance of \(x\) is given by \(\sigma^2 = \sum (x_i - \mu)^2 p_i\). Calculate \(\mu\) from Step 5, then use it to find \(\sigma^2\) using values from the table.

Compute the standard deviation

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

Find the cumulative distribution function

Calculate \(F(x)\) for all possible outcomes: \(F(x) = P(X \leq x)\). For example, \(F(-3) = P(X = -3)\), \(F(-1) = P(X \leq -1)\), etc.

Plot the cumulative distribution function

Using the calculated values from Step 8, plot \(F(x)\). The x-axis will be the possible values of \(x\) and the y-axis will be \(F(x)\). Connect the points to visualize the cumulative distribution function.

Key concepts

Sample Space

In probability, a sample space is the set of all possible outcomes of a random experiment. For this problem, we toss a weighted coin three times. Each toss can result in either heads (H) or tails (T). The sample space includes every possible sequence of these outcomes.

Here are all possible sequences when we toss the coin three times: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Each sequence is a sample point in our sample space. By analyzing these sequences, we can compute the values of the random variable and calculate their probabilities.

Mean Calculation

The mean, or expected value, of a random variable provides information about the center of its distribution. To find the mean, we use the formula \(\mu = \Sigma x_i p_i\), where \(x_i\) are the values of the random variable and \( p_i\) are the corresponding probabilities.

Using our weighted coin example, we have a table of values and probabilities:
- \(x = 3\), with probability \(p^3\)
- \(x = 1\), with probability \(3p^2(1-p)\)
- \(x = -1\), with probability \(3p(1-p)^2\)
- \(x = -3\), with probability \((1-p)^3\)

Now, substituting these into the mean formula gives us the expected value for the random variable.

Variance Calculation

The variance of a random variable measures the spread or dispersion of its values around the mean. The formula for variance is \(\sigma^2 = \Sigma (x_i - \mu)^2 p_i\).

Start by computing the mean \(\mu\) using the process from the previous section. Once you have \(\mu\), substitute the values of the random variable \(x_i\) and their probabilities \( p_i\) into the variance formula.

This tells us how much the actual values of the random variable differ from the mean, giving a sense of the variability in our data.

Standard Deviation Calculation

Standard deviation is another measure of the spread of a random variable's values. It is simply the square root of the variance. The formula is \(\sigma = \sqrt{\sigma^2}\).

After calculating the variance \(\sigma^2\), determine the standard deviation by taking its square root. This value reflects the average distance of the random variable's values from the mean, providing insight into overall variability in a more intuitive unit of measure than variance.

Cumulative Distribution Function

The cumulative distribution function (CDF) of a random variable describes the probability that the variable takes on a value less than or equal to a given number. To calculate the CDF \(F(x)\), sum the probabilities of all sample points less than or equal to \(x\).

Using our weighted coin example, calculate \(F(x)\) for all possible values of \(x\):
- \(F(-3) = P(X = -3)\)
- \(F(-1) = P(X \leq -1)\)
- \(F(1) = P(X \leq 1)\)
- \(F(3) = P(X \leq 3)\)

Once calculated, plot \(F(x)\) on a graph with the x-axis representing the values of \(x\) and the y-axis representing \(F(x)\). Connect the points to visualize the CDF.