/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 There are two traffic lights on ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 38

There are two traffic lights on my way to work. Let \(X_{1}\) be the number of lights at which I must stop, and suppose that the distribution of \(X_{1}\) is as follows: $$ \begin{array}{l|rrrr} x_{1} & 0 & 1 & 2 & \mu=1.1, \sigma^{2}=.49 \\ \hline p\left(x_{1}\right) & .2 & .5 & .3 \end{array} $$ Let \(X_{2}\) be the number of lights at which I must stop on the way home; \(X_{2}\) is independent of \(X_{1}\). Assume that \(X_{2}\) has the same distribution as \(X_{1}\), so that \(X_{1}, X_{2}\) is a random sample of size \(n=2\). a. Let \(T_{o}=X_{1}+X_{2}\), and determine the probability distribution of \(T_{o}\) b. Calculate \(\mu_{T_{e}}\). How does it relate to \(\mu\), the population mean? c. Calculate \(\sigma_{T_{0}}^{2}\). How does it relate to \(\sigma^{2}\), the population variance?

Short Answer

Expert verified
\( T_o \) distribution: 0, 0.04; 1, 0.2; 2, 0.37; 3, 0.3; 4, 0.09.\n\(\mu_{T_o} = 2.2, \sigma^2_{T_o} = 0.98\).

Step by step solution

01

Determine Values of T_o

The variable \( T_o = X_1 + X_2 \) can take values 0 through 4 since both \( X_1 \) and \( X_2 \) can be 0, 1, or 2. Add the possible values of \( X_1 \) and \( X_2 \) to find all possible combinations of \( T_o \): \(0+0, 0+1, 0+2, 1+0, 1+1, 1+2, 2+0, 2+1, 2+2\). These values are \(0, 1, 2, 3,\) and \(4\).
02

Calculate Distribution of T_o

List the combinations for each possible \( T_o \) value:- \( T_o = 0 \): \( (0,0) \) - \( T_o = 1 \): \((0,1), (1,0)\)- \( T_o = 2 \): \((0,2), (2,0), (1,1)\)- \( T_o = 3 \): \((1,2), (2,1)\)- \( T_o = 4 \): \((2,2)\).Use independence and the given probabilities:\( P(0,0) = 0.2 \times 0.2 = 0.04 \)\( P(0,1) = P(1,0) = 0.2 \times 0.5 = 0.1 \) \( \text{(each)} \)\( P(0,2) = P(2,0) = 0.2 \times 0.3 = 0.06 \) \( \text{(each)} \)\( P(1,1) = 0.5 \times 0.5 = 0.25 \)\( P(1,2) = P(2,1) = 0.5 \times 0.3 = 0.15 \) \( \text{(each)} \)\( P(2,2) = 0.3 \times 0.3 = 0.09 \).Sum these probabilities for each \( T_o \).Distribution:- \( T_o = 0 \), \( P(T_o = 0) = 0.04 \)- \( T_o = 1 \), \( P(T_o = 1) = 0.2 \)- \( T_o = 2 \), \( P(T_o = 2) = 0.37 \)- \( T_o = 3 \), \( P(T_o = 3) = 0.3 \)- \( T_o = 4 \), \( P(T_o = 4) = 0.09 \).
03

Calculate Mean of T_o (\mu_{T_o})

Using the property of the sum of independent random variables:\[ \mu_{T_o} = \mu_{X_1} + \mu_{X_2} = 1.1 + 1.1 = 2.2 \]
04

Calculate Variance of T_o (\sigma^2_{T_o})

Again using the sum of independent random variables:\[ \sigma^2_{T_o} = \sigma^2_{X_1} + \sigma^2_{X_2} = 0.49 + 0.49 = 0.98 \]
05

Conclusion Step

The computations show that the expected value \( \mu_{T_o} \) is simply twice the expected value \( \mu \) of one random variable, and the variance \( \sigma^2_{T_o} \) is also twice the variance \( \sigma^2 \) of one variable.

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Key Concepts

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

Random Variables
In probability and statistics, a random variable is a numerical outcome of a random process. It is like a box that can hold different outcomes from some event. The outcomes are not specific numbers but are associated with probabilities. For instance, consider the situation with the two traffic lights on your way to work. Here, the number of lights at which you must stop each day can vary. Therefore, we assign a random variable, like \( X_1 \), to represent this changing number.
  • **Discrete Random Variables**: These take specific values, like 0, 1, or 2 for \( X_1 \).
  • **Probability Distribution**: Shows how probabilities are spread over the values of a random variable, such as \( P(X_1 = 0) = 0.2 \).
Random variables provide a structured way to deal with uncertainty, turning randomness into a set of predictable outcomes.
Expected Value
The expected value, often denoted by \( \mu \), is a measure of the central tendency of a random variable. It is like the average outcome if you repeated the experiment many times. To find the expected value of a random variable, you multiply each possible value by its probability and then add all these products together. For example, with the random variable \( X_1 \) for the traffic lights, the expected value or mean, \( \mu_{X_1} \), is 1.1.
  • **Calculation**: For \( X_1 \), it's calculated as \( 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1 \).
  • **Importance**: It gives us an idea of what we expect to happen on average over time.
  • **Additivity**: If variables are independent, like \( X_1 \) and \( X_2 \), then the expected value of the sum is the sum of the expected values, as with \( T_o = X_1 + X_2 \) where \( \mu_{T_o} = 1.1 + 1.1 = 2.2 \).
Variance
Variance measures how much the values of a random variable differ from the mean, giving us a sense of the spread or "spread-out-ness" of the data. Think of it as telling you how much the outcomes fluctuate. A small variance means the values are close to the expected value, while a large variance indicates more spread.For the traffic lights, the variance of \( X_1 \) is 0.49.
  • **Calculation**: For \( X_1 \), variance \( \sigma^2 \) comes from \( (0-1.1)^2 \times 0.2 + (1-1.1)^2 \times 0.5 + (2-1.1)^2 \times 0.3 \).
  • **Additivity for Independent Variables**: The variance of the sum of two independent random variables is the sum of their variances, hence \( \sigma^2_{T_o} = 0.49 + 0.49 = 0.98 \).
Variance is crucial for understanding data variability, helping anticipate the level of surprises we might face in real-world applications.
Independent Events
Events are independent if the occurrence of one does not affect the occurrence of another. Independence simplifies calculations because each event can be considered in isolation. For our traffic lights problem, the number of stops on the way to work (\( X_1 \)) is independent of the number on the way home (\( X_2 \)).Several important things arise from independence:
  • **Probability Calculations**: If \( X_1 \) and \( X_2 \) are independent, the probability of them jointly taking specific values equals the product of their marginal probabilities, e.g., \( P(X_1 = 0, X_2 = 0) = P(X_1 = 0) \times P(X_2 = 0) \).
  • **Expected Values and Variances are Additive**: As shown with \( T_o \), the expected value of \( X_1 + X_2 \) equals the sum of their expected values, and similarly for variance.
Understanding independence allows for breaking down complex scenarios into simpler, solvable parts, making it easier to analyze probabilistic situations.

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Most popular questions from this chapter

Two airplanes are flying in the same direction in adjacent parallel corridors. At time \(t=0\), the first airplane is \(10 \mathrm{~km}\) ahead of the second one. Suppose the speed of the first plane \((\mathrm{km} / \mathrm{hr})\) is normally distributed with mean 520 and standard deviation 10 and the second plane's speed is also normally distributed with mean and standard deviation 500 and 10 , respectively. a. What is the probability that after \(2 \mathrm{hr}\) of flying, the second plane has not caught up to the first plane? b. Determine the probability that the planes are separated by at most \(10 \mathrm{~km}\) after \(2 \mathrm{hr}\).

An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let \(X=\) the \(x\) coordinate of the point selected and \(Y=\) the \(y\) coordinate of the point selected. If the circle is centered at \((0,0)\) and has radius \(R\), then the joint pdf of \(X\) and \(Y\) is $$ f(x, y)=\left\\{\begin{array}{cl} \frac{1}{\pi R^{2}} & x^{2}+y^{2} \leq R^{2} \\ 0 & \text { otherwise } \end{array}\right. $$ a. What is the probability that the selected point is within \(R / 2\) of the center of the circular region? [Hint: Draw a picture of the region of positive density D. Because \(f(x, y)\) is constant on \(D\), computing a probability reduces to computing an area.] b. What is the probability that both \(X\) and \(Y\) differ from 0 by at most \(R / 2\) ? c. Answer part (b) for \(R / \sqrt{2}\) replacing \(R / 2\). d. What is the marginal pdf of \(X\) ? Of \(Y\) ? Are \(X\) and \(Y\) independent?

a. Use the rules of expected value to show that \(\operatorname{Cov}(a X+b\), \(c Y+d)=a c \operatorname{Cov}(X, Y)\). b. Use part (a) along with the rules of variance and standard deviation to show that \(\operatorname{Corr}(a X+b, c Y+d)=\operatorname{Corr}(X, Y)\) when \(a\) and \(c\) have the same sign. c. What happens if \(a\) and \(c\) have opposite signs?

Five automobiles of the same type are to be driven on a 300-mile trip. The first two will use an economy brand of gasoline, and the other three will use a name brand. Let \(X_{1}, X_{2}\), \(X_{3}, X_{4}\), and \(X_{5}\) be the observed fuel efficiencies (mpg) for the five cars. Suppose these variables are independent and normally distributed with \(\mu_{1}=\mu_{2}=20, \mu_{3}=\mu_{4}=\mu_{5}=21\), and \(\sigma^{2}=4\) for the economy brand and \(3.5\) for the name brand. Define an rv \(Y\) by $$ Y=\frac{X_{1}+X_{2}}{2}-\frac{X_{3}+X_{4}+X_{5}}{3} $$ so that \(Y\) is a measure of the difference in efficiency between economy gas and name-brand gas. Compute \(P(0 \leq Y)\) and \(P(-1 \leq Y \leq 1) .\left[\right.\)

A particular brand of dishwasher soap is sold in three sizes: \(25 \mathrm{oz}, 40 \mathrm{oz}\), and \(65 \mathrm{oz}\). Twenty percent of all purchasers select a 25 -oz box, \(50 \%\) select a 40 -oz box, and the remaining \(30 \%\) choose a \(65-\mathrm{oz}\) box. Let \(X_{1}\) and \(X_{2}\) denote the package sizes selected by two independently selected purchasers. a. Determine the sampling distribution of \(\bar{X}\), calculate \(E(\bar{X})\), and compare to \(\mu\). b. Determine the sampling distribution of the sample variance \(S^{2}\), calculate \(E\left(S^{2}\right)\), and compare to \(\sigma^{2}\).
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