91Ó°ÊÓ

Combination of Random Variables: Golf Norb and Gary are entered in a local golf tournament. Both have played the local course many times. Their scores are random variables with the following means and standard deviations. $$ \text { Norb, } x_{1}: \mu_{1}=115 ; \sigma_{1}=12 \quad \text { Gary, } x_{2}: \mu_{2}=100 ; \sigma_{2}=8 $$ In the tournament, Norb and Gary are not playing together, and we will assume their scores vary independently of each other. (a) The difference between their scores is \(W=x_{1}-x_{2} .\) Compute the mean, variance, and standard deviation for the random variable \(W\). (b) The average of their scores is \(W=0.5 x_{1}+0.5 x_{2}\). Compute the mean, variance, and standard deviation for the random variable \(W\). (c) The tournament rules have a special handicap system for each player. For Norb, the handicap formula is \(L=0.8 x_{1}-2 .\) Compute the mean, variance, and standard deviation for the random variable \(L .\) (d) For Gary, the handicap formula is \(L=0.95 x_{2}-5 .\) Compute the mean, variance, and standard deviation for the random variable \(L .\)

Step by step solution

01

Identify Parameters for Norb and Gary

Given the means and standard deviations: Norb has a mean score \(\mu_1 = 115\) and a standard deviation \(\sigma_1 = 12\). Gary has a mean score \(\mu_2 = 100\) and a standard deviation \(\sigma_2 = 8\). Their scores are independent random variables.

02

Compute Mean of the Difference in Scores

The mean of the difference \(W = x_1 - x_2\) is the difference of the means: \( \mu_W = \mu_1 - \mu_2 = 115 - 100 = 15 \).

03

Compute Variance of the Difference in Scores

Since the scores are independent, \( \text{Var}(W) = \text{Var}(x_1) + \text{Var}(x_2) = \sigma_1^2 + \sigma_2^2 = 12^2 + 8^2 = 144 + 64 = 208 \).

04

Compute Standard Deviation of the Difference in Scores

The standard deviation is the square root of the variance: \( \sigma_W = \sqrt{208} \approx 14.42 \).

05

Compute Mean of the Average Scores

The mean of \(W = 0.5 x_1 + 0.5 x_2\) is \( \mu_W = 0.5\mu_1 + 0.5\mu_2 = 0.5(115) + 0.5(100) = 57.5 + 50 = 107.5 \).

06

Compute Variance of the Average Scores

\( \text{Var}(W) = (0.5^2)\text{Var}(x_1) + (0.5^2)\text{Var}(x_2) = 0.25(144) + 0.25(64) = 36 + 16 = 52 \).

07

Compute Standard Deviation of the Average Scores

The standard deviation of the average is \( \sigma_W = \sqrt{52} \approx 7.21 \).

08

Compute Mean for Norb's Handicap

For \(L = 0.8 x_1 - 2\), the mean is \( \mu_L = 0.8\mu_1 - 2 = 0.8(115) - 2 = 92 - 2 = 90 \).

09

Compute Variance and Standard Deviation for Norb's Handicap

\( \text{Var}(L) = (0.8^2)\text{Var}(x_1) = 0.64(144) = 92.16 \). The standard deviation \( \sigma_L = \sqrt{92.16} \approx 9.60 \).

10

Compute Mean for Gary's Handicap

For \(L = 0.95 x_2 - 5\), the mean is \( \mu_L = 0.95\mu_2 - 5 = 0.95(100) - 5 = 95 - 5 = 90 \).

11

Compute Variance and Standard Deviation for Gary's Handicap

\( \text{Var}(L) = (0.95^2)\text{Var}(x_2) = 0.9025(64) = 57.76 \). The standard deviation \( \sigma_L = \sqrt{57.76} \approx 7.60 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Variables

A random variable is a key concept in statistics, particularly when studying phenomena that have inherent randomness, like our golfers' scores. It is essentially a variable whose possible values are numerical outcomes of a random phenomenon. For instance, when considering Norb and Gary's scores, each score is a random variable because it can vary from one game to another.

Random variables come in two types: discrete and continuous. Discrete random variables take on a countable number of distinct values. An example would be the number of goals in a soccer match. Continuous random variables, like our golfers' scores, can take infinitely many values within a given range.

Understanding random variables allows us to model uncertainty and make predictions based on observed data. In our exercise, each golfer's score is treated as a continuous random variable, which means their scores can take any value from a range, reflecting their performance variability.

Mean and Variance

The mean and variance are two fundamental statistics that provide significant insights into the nature of a random variable.

• Mean (Expected Value): This represents the average or expected value of the random variable. For example, Norb's average score (mean) is 115, and Gary's is 100. It provides a central value around which the scores tend to cluster.
• Variance: This measures how much the values of the random variable spread out around the mean. The larger the variance, the more spread out the scores are. In our exercise, Norb's score variance is the square of his standard deviation, which is 144, and Gary's is 64.

Calculating the mean involves summing up all possible values of the random variable, each multiplied by its probability of occurrence. Variance is calculated as the average of the squared deviations from the mean, which helps to understand the degree of dispersion in the data.

Standard Deviation

Standard deviation is an extension of variance and is one of the most widely used measures of dispersion in statistical data.

• What it Represents: It represents the average amount by which the values of a random variable differ from the mean. A smaller standard deviation means the data points are closer to the mean, whereas a larger one indicates they are more spread out.
• Calculation: The standard deviation is simply the square root of the variance. For instance, with a variance of 208 for the difference between Norb and Gary's scores, the standard deviation is approximately 14.42.

Standard deviation is extremely useful because it gives us an indication of the "typical" deviation from the mean in units of the variable. It allows statisticians and researchers to comment on the reliability and accuracy of the data, making it a key concept in many fields of study.

Handicap System

Handicap systems are used in various sports to ensure that players of different skill levels can compete on a more equal footing. In golf, the handicap reflects a player's average performance, adjusting scores to promote fairness.

In our example, both Norb and Gary have handicap calculations that adjust their scores. Norb's formula is given by: \(L = 0.8 x_1 - 2\), while Gary's is \(L = 0.95 x_2 - 5\). These formulas adjust their actual scores to account for their playing ability, enabling comparisons with other players more equitably.

This adjustment impacts the mean and variance of their scores. By applying the handicap formula, the mean, variance, and standard deviation of the adjusted scores can be calculated, illustrating how each player's performance is normalized. Such systems are crucial for maintaining competitiveness and enjoyment in sports events where participants have varying skill levels.