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All horses are the same color; we can prove this by induction on the number of horses in a given set. Here's how: "If there's just one horse then it's the same color as itself, so the basis is trivial. For the induction step, assume that there are $n$ horses numbered 1 to $n$. By the induction hypothesis, horses 1 through $n-1$ are the same color, and similarly horses 2 through $n$ are the same color. But the middle horses, 2 through $n-1$, can't change color when they're in different groups; these are horses, not chameleons. So horses 1 and $n$ must be the same color as well, by transitivity. Thus all $\mathrm{n}$ horses are the same color; QED." What, if anything, is wrong with this reasoning?

All horses are the same color; we can prove this by induction on the number of horses in a given set. Here's how: "If there's just one horse then it's the same color as itself, so the basis is trivial. For the induction step, assume that there are $n$ horses numbered 1 to $n$. By the induction hypothesis, horses 1 through $n-1$ are the same color, and similarly horses 2 through $n$ are the same color. But the middle horses, 2 through $n-1$, can't change color when they're in different groups; these are horses, not chameleons. So horses 1 and $n$ must be the same color as well, by transitivity. Thus all $\mathrm{n}$ horses are the same color; QED." What, if anything, is wrong with this reasoning?

Concrete Mathematics: A Foundation for Computer Science

Which of the following binary operations are closed? a. subtraction of positive integers b. division of nonzero integers c. function composition of polynomials with real coefficients d. multiplication of $2 \times 2$ matrices with integer entries

Contemporary Abstract Algebra

Let $\mathbf{R}^{+}$ be the group of positive real numbers under multiplication. Show that the mapping $\phi(x)=\sqrt{x}$ is an automorphism of $\mathbf{R}^{+}$.

Contemporary Abstract Algebra

For $n=5,8,12,20$, and 25, find all positive integers less than $n$ and relatively prime to $n$.

Contemporary Abstract Algebra

Questions asked

ANSWERED

Jacob Fry verified

Numerade educator

Let ( Y_{1} ) and ( Y_{2} ) be uncorrelated random variables and consider ( U_{1}=Y_{1}+Y_{2} ) and ( U_{2}=Y_{1}-Y_{2} ). a Find the ( operatorname{Cov}left(U_{1}, U_{2} ight) ) in terms of the variances of ( Y_{1} ) and ( Y_{2} ). b Find an expression for the coefficient of correlation between ( U_{1} ) and ( U_{2} ). c Is it possible that ( operatorname{Cov}left(U_{1}, U_{2} ight)=0 ) ? When does this occur?

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ANSWERED

Rashmi Sinha verified

Numerade educator

Let ?1 and ?2 be uncorrelated random variables and consider ?1 = ?1 + ?2 and ?2 = ?1 − ?2. a. Find the covariance Cov(?1 , ?2) in terms of the variances of ?1 and ?2. b. Find an expression for the coefficient of correlation between ?1 and ?2. c. Is it possible that Cov(?1 , ?2) = 0? When does this occur?

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ANSWERED

Kaushal Nair verified

Numerade educator

Let X and Y be two random variables with joint density function ?(?, ?) = ? + ? if 0 ≤ ?, ? ≤ 1, zero elsewhere. Find ?(? < 2?).

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INSTANT ANSWER

The joint density function of \( Y_{1} \) and \( Y_{2} \) is given by \[ f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} 30 y_{1} y_{2}^{2}, & y_{1}-1 \leq y_{2} \leq 1-y_{1}, 0 \leq y_{1} \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \] a Find \( F(1 / 2,1 / 2) \). b Find \( F(1 / 2,2) \). c Find \( P\left(Y_{1}>Y_{2}\right) \).

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INSTANT ANSWER

The joint density function of \( Y_{1} \) and \( Y_{2} \) is given by \[ f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} 30 y_{1} y_{2}^{2}, & y_{1}-1 \leq y_{2} \leq 1-y_{1}, 0 \leq y_{1} \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \] a Find \( F(1 / 2,1 / 2) \). b Find \( F(1 / 2,2) \). c Find \( P\left(Y_{1}>Y_{2}\right) \).

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INSTANT ANSWER

An electronic system has one each of two different types of components in joint operation. Let \( y_{1} \) and \( y_{2} \) denote the random lengths of life of the components of type 1 and type II, respectively. The joint density function is given by \[ f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} (1 / 8) y_{1} e^{-\left(y_{1}+y_{1}\right) / 2}, & y_{1}>0, y_{2}>0 \\ 0, & \text { elsewhere } \end{array}\right. \] (Measurements are in hundreds of hours.) Find \( P_{\left(Y_{1}>1, Y_{2}>1\right)} \).

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ANSWERED

Rashmi Sinha verified

Numerade educator

Suppose that Y? and Y? are uniformly distributed over the triangle shaded in the accompanying diagram. a. Find P(Y? ? 3/4, Y? ? 3/4). b. Find P(Y? - Y? ? 0).

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INSTANT ANSWER

Let \( Y_{1} \) and \( Y_{2} \) have the joint probability density function given by \[ f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} k y_{1} y_{2}, & 0 \leq y_{1} \leq 1,0 \leq y_{2} \leq 1 \\ 0, & \text { elsewhere } \end{array}\right. \] a. Find the value of \( { }_{k} \) that makes this a probability density function. b. Find the joint distribution function for \( r_{1} \) and \( r_{2} \). c. Find \( P\left(Y_{1} \leq 1 / 2, Y_{2} \leq 3 / 4\right) \).

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INSTANT ANSWER

Given here is the joint probability function associated with data obtained in a study of automobile accidents in which a child (under age 5 years) was in the car and at least one fatality occurred. Specifically, the study focused on whether or not the child survived and what type of seatbelt (if any) he or she used. Define \[ Y_{1}=\left\{\begin{array}{ll} 0, & \text { if the child surrived, } \\ 1, \text { if not } \end{array}\right. \] Notice that \( y_{1} \) is the number of fatalities per child and, since children's car seats usually utilize two belts, \( v_{2} \) is the number of seatbelts in use at the time of the accident. a. Verify that the preceding probability function satisfies Theorem 5.1. b. Find \( F_{F(1,2)} \). What is the interpretation of this value?

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ANSWERED

Jacob Fry verified

Numerade educator

The relative humidity ( Y ), when measured at a location, has a probability density function given by [ f(y)=left{egin{array}{ll} k y^{3}(1-y)^{2}, & 0 leq y leq 1, \ 0, & ext { elsewhere } end{array} ight. ] a Find the value of ( k ) that makes ( f(y) ) a density function. b find a humidity value that is exceeded only ( 5 % ) of the time.

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