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Chapter 15: Problem 32

Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of: $$\begin{array}{c|c|c} & \text { Mean } & \text { SD } \\\\\hline X & 80 & 12 \\\Y & 12 & 3\end{array}$$ a) \(2 Y+20\) b) \(3 X\) c) \(0.25 X+Y\) d) \(X-5 Y\) e) \(X_{1}+X_{2}+X_{3}\)

Short Answer

Expert verified
a) Mean = 44, SD = 6. b) Mean = 240, SD = 36. c) Mean = 32, SD = 4.24. d) Mean = 20, SD = 19.2. e) Mean = 240, SD = 20.8.

Step by step solution

01

Understand Transformation Impact on Mean

For any constant transformation of a random variable, such as multiplying or adding a constant, the mean is affected in a straightforward way. If a random variable \(Z = aY + b\), then \(E(Z) = aE(Y) + b\). Apply this to find the means.
02

Calculate Mean for Each Expression

a) For \(2Y + 20\), the mean is \(2 \times 12 + 20 = 44.\)b) For \(3X\), the mean is \(3 \times 80 = 240.\)c) For \(0.25X + Y\), the mean is \(0.25 \times 80 + 12 = 32.\)d) For \(X - 5Y\), the mean is \(80 - 5 \times 12 = 20.\)e) For \(X_1 + X_2 + X_3\), the mean is \(3 \times 80 = 240.\) as each \(X_i\) has the same mean.
03

Understand Transformation Impact on Standard Deviation

The standard deviation of a linear transformation \(Z = aY + b\) is given by \(\text{SD}(Z) = |a| \times \text{SD}(Y)\). Constants added or subtracted do not affect the standard deviation.
04

Calculate Standard Deviation for Each Expression

a) For \(2Y + 20\), the standard deviation is \(2 \times 3 = 6.\)b) For \(3X\), the standard deviation is \(3 \times 12 = 36.\)c) For \(0.25X + Y\), use \(\text{SD}(0.25X + Y) = \sqrt{(0.25 \times 12)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 4.24.\)d) For \(X - 5Y\), the standard deviation is \(\sqrt{12^2 + (5 \times 3)^2} = \sqrt{144 + 225} = \sqrt{369} = 19.2.\)e) For \(X_1 + X_2 + X_3\), the standard deviation is \(\sqrt{3} \times 12 = 20.8.\) since the standard deviations add in quadrature.

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Key Concepts

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

Mean Calculation
Mean calculation is a fundamental concept when working with random variables. The mean of a random variable provides a measure of its 'central' or 'average' value. When we perform operations such as scaling (multiplying by a constant) or translating (adding a constant) on a random variable, its mean is transformed accordingly.
To calculate the mean of a linear transformation, we use the formula:
  • For a random variable transformed by a formula like \(Z = aY + b\), the mean \(E(Z)\) is given by \(aE(Y) + b\).
  • This means you can directly multiply the coefficient by the mean and add any constant shift to get the new mean.

Applying this understanding, the mean for expressions like \(2Y + 20\) or \(3X\) can be easily found by performing these simple calculations using their respective means, as shown in the provided exercise solution.
Standard Deviation
The standard deviation measures the amount of variation or dispersion of a set of values. It is crucial in understanding how much the data points differ from the mean. Like the mean, when transformations are applied, the standard deviation is adjusted based on specific rules.
For linear transformations of the form \(Z = aY + b\), the standard deviation is calculated as:
  • \(\text{SD}(Z) = |a| \times \text{SD}(Y)\). This formula shows that only the scaling factor affects the standard deviation, while adding or subtracting a constant value does not.
  • For independent variables, if using \(X\) and \(Y\) together, we'd sum their variances before taking the square root, utilizing the relation for independent variables: \(\text{SD}(A + B) = \sqrt{\text{Var}(A) + \text{Var}(B)}\).
Understanding how the standard deviation changes with these operations helps in correctly interpreting the variability of transformed variables.
Linear Transformations
A linear transformation on a random variable involves operations like scaling by a factor or shifting via addition. These transformations have predictable effects on the mean and standard deviation of the random variable.
In general:
  • Multiplying a random variable by a constant \(a\) scales its mean and standard deviation by \(a\).
  • Adding or subtracting a constant \(b\) affects only the mean, leaving the standard deviation unchanged.
This understanding is vital when examining transformations like \(2Y + 20\), as both mean and standard deviation are adjusted according to these rules. Recognizing the transformations helps solve a variety of statistical problems quickly and accurately.
Independent Variables
Independent variables are those which do not influence each other in terms of their variations. This concept is particularly useful in probability and statistics when dealing with multiple random variables.
When manipulating such variables:
  • For mean calculations, simply add the means of the independent variables.
  • For standard deviations, you'll need to consider the variances (the square of the standard deviations) of each variable and sum them to get the variance of their sum. Then, take the square root to find the standard deviation of their combined outcome.

These principles allow you to handle complex operations involving multiple variables, such as finding the mean and standard deviation of \(X - 5Y\) or \(X_1 + X_2 + X_3\), efficiently.

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Most popular questions from this chapter

A company selling vegetable seeds in packets of 20 estimates that the mean number of seeds that will actually grow is \(18,\) with a standard deviation of 1.2 seeds. You buy 5 different seed packets. a) How many good seeds do you expect to get? b) What's the standard deviation? c) What assumptions did you make about the seeds? Do you think that assumption is warranted? Explain.

You draw a card from a deck. If you get a red card, you win nothing. If you get a spade, you win \(\$ 5 .\) For any club, you win \(\$ 10\) plus an extra \(\$ 20\) for the ace of clubs. a) Create a probability model for the amount you win. b) Find the expected amount you'll win. c) What would you be willing to pay to play this game?

A couple plans to have children until they get a girl, but they agree that they will not have more than three children even if all are boys. (Assume boys and girls are equally likely.) a) Create a probability model for the number of children they might have. b) Find the expected number of children. c) Find the expected number of boys they鈥檒l have.

A delivery company's trucks occasionally get parking tickets, and based on past experience, the company plans that the trucks will average 1.3 tickets a month, with a standard deviation of 0.7 tickets. a) If they have 18 trucks, what are the mean and standard deviation of the total number of parking tickets the company will have to pay this month? b) What assumption did you make in answering?

1 is 0.8. If you get contract \(\\# 1,\) the probability you also get contract \(\\# 2\) wil鈥� # Your company bids for two contracts. You believe the probability you get contract #1 is 0.8. If you get contract \(\\# 1,\) the probability you also get contract \(\\# 2\) will be \(0.2,\) and if you do not get \(\\# 1,\) the probability you get #2 will be 0.3. a) Are the two contracts independent? Explain. b) Find the probability you get both contracts. c) Find the probability you get no contract. d) Let \(X\) be the number of contracts you get. Find the probability model for \(X\) e) Find the expected value and standard deviation.
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