A consumer organization estimates that over a 1-year period 20% of cars will need to be repaired once, 6% will need repairs twice, and 2% will require three or more repairs. What is the probability that a car chosen at random will need repairs?

enmobladatn
2022-07-19
Answered

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salumeqi

Answered 2022-07-20
Author has **15** answers

In one year period

Probability of a randomly chosen car to be repaired once $=20\mathrm{\%}=\frac{20}{100}=0.2$

Probability of a randomly chosen car not to be repaired once =1--.2=0.8

Probability of a randomly chosen car to be repaired twice $=6\mathrm{\%}=\frac{6}{100}=0.06$

Probability of a randomly chosen car not to be repaired twice =1-0.06 =0.94

Probability of a randomly chosen car to be repaired three or more

$=2\mathrm{\%}=\frac{2}{100}=0.02$

Probability of a randomly chosen car not to be repaired three or more

=1-0.02=0.98

Probability of a randomly chosen car not to be repaired

$=0.2\times 0.94\times 0.98=0.18424$

Probability of a randomly chosen car need to be repaired = 1 - Probability of a randomly chosen car not to be repaired

=1-0.18424=0.81576

Probability of a randomly chosen car to be repaired once $=20\mathrm{\%}=\frac{20}{100}=0.2$

Probability of a randomly chosen car not to be repaired once =1--.2=0.8

Probability of a randomly chosen car to be repaired twice $=6\mathrm{\%}=\frac{6}{100}=0.06$

Probability of a randomly chosen car not to be repaired twice =1-0.06 =0.94

Probability of a randomly chosen car to be repaired three or more

$=2\mathrm{\%}=\frac{2}{100}=0.02$

Probability of a randomly chosen car not to be repaired three or more

=1-0.02=0.98

Probability of a randomly chosen car not to be repaired

$=0.2\times 0.94\times 0.98=0.18424$

Probability of a randomly chosen car need to be repaired = 1 - Probability of a randomly chosen car not to be repaired

=1-0.18424=0.81576

asked 2022-05-01

Suppose you have two sequences of complex numbers ${a}_{i}$ and ${b}_{i}$ indexed over the integer numbers such that they are convergent in ${l}^{2}$ norm and a has norm greater than b in the sense

$\mathrm{\infty}>\sum _{i}|{a}_{i}{|}^{2}\ge \sum _{i}|{b}_{i}{|}^{2}.$

Suppose moreover they are uncorrelated over any time delay, meaning

$\sum _{i}{a}_{i}\overline{{b}_{i-n}}=0\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\in \mathbb{Z}.$

Is it true that the polinomial $a(z)=\sum _{i}{a}_{i}{z}^{-i}$ is greater in absolute value than $b(z)=\sum _{i}{b}_{i}{z}^{-i}$ for any unit norm complex number z?

$\mathrm{\infty}>\sum _{i}|{a}_{i}{|}^{2}\ge \sum _{i}|{b}_{i}{|}^{2}.$

Suppose moreover they are uncorrelated over any time delay, meaning

$\sum _{i}{a}_{i}\overline{{b}_{i-n}}=0\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\in \mathbb{Z}.$

Is it true that the polinomial $a(z)=\sum _{i}{a}_{i}{z}^{-i}$ is greater in absolute value than $b(z)=\sum _{i}{b}_{i}{z}^{-i}$ for any unit norm complex number z?

asked 2022-06-24

Let a sample $(x,y)\in {\mathbb{R}}^{2n}$ be given, where $y$ only attains the values $0$ and $1$. We can try to model this data set by either linear regression

${y}_{i}={\alpha}_{0}+{\beta}_{0}{x}_{i}$

with the coefficients determined by the method of least squares or by logistic regression

${\pi}_{i}=\frac{\mathrm{exp}({\alpha}_{1}+{\beta}_{1}{x}_{i})}{1+\mathrm{exp}({\alpha}_{1}+{\beta}_{1}{x}_{i})},$

where ${\pi}_{i}$ denotes the probability that ${y}_{i}=1$ under the given value ${x}_{i}$ and the coefficients are determined by the Maximum-Likelihood method. My question is whether the following statement holds true.

Claim: If ${\beta}_{0}>0$ (${\beta}_{0}<0$), then ${\beta}_{1}>0$ (${\beta}_{1}>0$).

I figure this could be due to the sign of the correlation coefficient.

${y}_{i}={\alpha}_{0}+{\beta}_{0}{x}_{i}$

with the coefficients determined by the method of least squares or by logistic regression

${\pi}_{i}=\frac{\mathrm{exp}({\alpha}_{1}+{\beta}_{1}{x}_{i})}{1+\mathrm{exp}({\alpha}_{1}+{\beta}_{1}{x}_{i})},$

where ${\pi}_{i}$ denotes the probability that ${y}_{i}=1$ under the given value ${x}_{i}$ and the coefficients are determined by the Maximum-Likelihood method. My question is whether the following statement holds true.

Claim: If ${\beta}_{0}>0$ (${\beta}_{0}<0$), then ${\beta}_{1}>0$ (${\beta}_{1}>0$).

I figure this could be due to the sign of the correlation coefficient.

asked 2022-05-24

Is there a method for polynomial regression in $$2D$$ dimensions (fitting a function $$f(x,y)$$ to a set of data $$X,Y$$, and $$Z$$)? And is there a way to apply a condition to the regression in $$2D$$ that requires all functions fitted to go through the axis line $$x=0$$?

asked 2022-07-16

The process to generate the two RVs is as follows. We first draw T from $Uniform(0,1)$ . If $T\le 0.5$ we take $X=T$ and draw Y from $Uniform(0,1)$ . Otherwise if $T>0.5$ , we take $Y=T$ and draw X from $Uniform(0,1)$ . Running a simulation it seems like X and Y are positively correlated, though intuitively it seems like they should have no effect on each other. What is the explanation?

asked 2022-07-20

a. Two variables have a positive linear correlation. Does the dependent variable increase or decrease as the independent variable increases? Explain.

b. Two variables have a negative linear correlation. Does the dependent variable increase or decrease as the independent variable increases?

c. Describe the range of values for the correlation coefficient, r.

d. What does the sample correlation coefficient r measure? Which value indicates a stronger correlation:

${r}_{1}=0.975$ or ${r}_{2}=-0.987$. Explain, please.

b. Two variables have a negative linear correlation. Does the dependent variable increase or decrease as the independent variable increases?

c. Describe the range of values for the correlation coefficient, r.

d. What does the sample correlation coefficient r measure? Which value indicates a stronger correlation:

${r}_{1}=0.975$ or ${r}_{2}=-0.987$. Explain, please.

asked 2022-07-18

Multiple regression problems (restricted regression, dummy variables)

Q1.

Model 1: $Y={X}_{1}{\beta}_{1}+\epsilon $

Model 2: $Y={X}_{1}{\beta}_{1}+{X}_{2}{\beta}_{2}+\epsilon $

(a) Suppose that Model 1 is true. If we estimates OLS estrimator ${b}_{1}$ for ${\beta}_{1}$ in Model 2, what will happen to the size and power properties of the test?

(b) Suppose that Model 2 is true. If we estimates OLS estrimator ${b}_{1}$ for ${\beta}_{1}$ in Model 1, what will happen to the size and power properties of the test?

-> Here is my guess.

(a) ${b}_{1}$ is unbiased, inefficient estimator. (I calculated it using formula for "inclusion of irrelevant variable" and ${b}_{1}=({X}_{1}^{\prime}{M}_{2}{X}_{1}{)}^{-1}{X}_{1}^{\prime}{M}_{2}Y$ where ${M}_{2}$ is symmetric and idempotent matrix) Inefficient means that it has larger variance thus size increases and power increases too.

(b) ${b}_{1}$ is biased, efficient estimator. (I use formular for "exclusion of relevant variable" and ${b}_{1}=({X}_{1}^{\prime}{X}_{1}{)}^{-1}{X}_{1}^{\prime}Y$) Um... I stuck here. What should I say using that information?

Q2.

Let $Q$ and $P$ be the quantity and price. Relation between them is different across reions of east, west, south and north, and as well, for different 4 seasons. Construct a model.

-> Actually, I don't know well about dummy variables. So any please solve this problem to help me.

Q1.

Model 1: $Y={X}_{1}{\beta}_{1}+\epsilon $

Model 2: $Y={X}_{1}{\beta}_{1}+{X}_{2}{\beta}_{2}+\epsilon $

(a) Suppose that Model 1 is true. If we estimates OLS estrimator ${b}_{1}$ for ${\beta}_{1}$ in Model 2, what will happen to the size and power properties of the test?

(b) Suppose that Model 2 is true. If we estimates OLS estrimator ${b}_{1}$ for ${\beta}_{1}$ in Model 1, what will happen to the size and power properties of the test?

-> Here is my guess.

(a) ${b}_{1}$ is unbiased, inefficient estimator. (I calculated it using formula for "inclusion of irrelevant variable" and ${b}_{1}=({X}_{1}^{\prime}{M}_{2}{X}_{1}{)}^{-1}{X}_{1}^{\prime}{M}_{2}Y$ where ${M}_{2}$ is symmetric and idempotent matrix) Inefficient means that it has larger variance thus size increases and power increases too.

(b) ${b}_{1}$ is biased, efficient estimator. (I use formular for "exclusion of relevant variable" and ${b}_{1}=({X}_{1}^{\prime}{X}_{1}{)}^{-1}{X}_{1}^{\prime}Y$) Um... I stuck here. What should I say using that information?

Q2.

Let $Q$ and $P$ be the quantity and price. Relation between them is different across reions of east, west, south and north, and as well, for different 4 seasons. Construct a model.

-> Actually, I don't know well about dummy variables. So any please solve this problem to help me.

asked 2022-07-03

Let the joint distribution of (X, Y) be bivariate normal with mean vector $\left(\begin{array}{c}0\\ 0\end{array}\right)$ and variance-covariance matrix

$\left(\begin{array}{cc}1& \bm{\rho}\\ \bm{\rho}& 1\end{array}\right)$ , where $-\U0001d7cf<\bm{\rho}<\U0001d7cf$ . Let ${\mathbf{\Phi}}_{\bm{\rho}}(\U0001d7ce,\U0001d7ce)=\bm{P}(\bm{X}\le \U0001d7ce,\bm{Y}\le \U0001d7ce)$ . Then what will be Kendall’s $\tau $ coefficient between X and Y equal to?

$\left(\begin{array}{cc}1& \bm{\rho}\\ \bm{\rho}& 1\end{array}\right)$ , where $-\U0001d7cf<\bm{\rho}<\U0001d7cf$ . Let ${\mathbf{\Phi}}_{\bm{\rho}}(\U0001d7ce,\U0001d7ce)=\bm{P}(\bm{X}\le \U0001d7ce,\bm{Y}\le \U0001d7ce)$ . Then what will be Kendall’s $\tau $ coefficient between X and Y equal to?