True or false: if ${a}_{n}$ is any decreasing sequence of positive real numbers and ${b}_{n}$ is any sequence of real numbers converges to $0$, then $\frac{{a}_{n}}{{b}_{n}}$ diverges.

velinariojepvg
2022-05-09
Answered

True or false: if ${a}_{n}$ is any decreasing sequence of positive real numbers and ${b}_{n}$ is any sequence of real numbers converges to $0$, then $\frac{{a}_{n}}{{b}_{n}}$ diverges.

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Maeve Holloway

Answered 2022-05-10
Author has **23** answers

if ${a}_{n}={e}^{-n}$ and ${b}_{n}=\frac{1}{{n}^{2}}$ then

$({a}_{n})$ is a decreasing sequence of positive numbers and $\underset{+\mathrm{\infty}}{lim}{b}_{n}=0$

but

$\frac{{a}_{n}}{{b}_{n}}={n}^{2}{e}^{-n}\to 0.$

$({a}_{n})$ is a decreasing sequence of positive numbers and $\underset{+\mathrm{\infty}}{lim}{b}_{n}=0$

but

$\frac{{a}_{n}}{{b}_{n}}={n}^{2}{e}^{-n}\to 0.$

Jaeden Weaver

Answered 2022-05-11
Author has **3** answers

Hint: Take ${a}_{n}={b}_{n}=\frac{1}{n}$

asked 2022-04-07

The prevalence of breast cancer in women over 40 in country X is estimated to be 0.8% (i.e., 8 in every 1,000 women in that age group).

Mammograms test for the presence of breast cancer. A positive result indicates that the disease is present. A negative result indicates that it is not.

The sensitivity of a mammogram test for breast cancer is estimated to be 90%. This is the probability that the mammogram will give a positive result when the person being tested does have breast cancer.

The false positive rate for the mammogram is 7.5%. This is the probability that the mammogram will give a positive test result when the person being tested does not have breast cancer.

All women who test positive (816) in the mammogram are referred for a further, different examination, which however has the same sensitivity and false positive rates as the first test.

What is the probability that a woman referred for this examination and testing positive again, actually does have breast cancer?

Mammograms test for the presence of breast cancer. A positive result indicates that the disease is present. A negative result indicates that it is not.

The sensitivity of a mammogram test for breast cancer is estimated to be 90%. This is the probability that the mammogram will give a positive result when the person being tested does have breast cancer.

The false positive rate for the mammogram is 7.5%. This is the probability that the mammogram will give a positive test result when the person being tested does not have breast cancer.

All women who test positive (816) in the mammogram are referred for a further, different examination, which however has the same sensitivity and false positive rates as the first test.

What is the probability that a woman referred for this examination and testing positive again, actually does have breast cancer?

asked 2022-05-10

in order to remember stuff i need to understand their reason. Right now i cannot remember what is type 1 error and what is type 2 error why is the reason type 1 is false positive?

asked 2022-05-07

How the false positive value affects accuracy?

$TP\text{}(true\text{}positive)\text{}=\text{}2739$

$TN\text{}(true\text{}negative)\text{}=\text{}103217$

$FP\text{}(false\text{}positive)\text{}=\text{}43423$

$FN\text{}(false\text{}negative)\text{}=\text{}5022$

$accuracy=\frac{TP+TN}{TP+TN+FP+FN}$

In this case the accuracy is $0.68$. Can I say that I have low accuracy because the value false positive is high? There is any relathion between false positive and the parameters true positive or true negative?

$TP\text{}(true\text{}positive)\text{}=\text{}2739$

$TN\text{}(true\text{}negative)\text{}=\text{}103217$

$FP\text{}(false\text{}positive)\text{}=\text{}43423$

$FN\text{}(false\text{}negative)\text{}=\text{}5022$

$accuracy=\frac{TP+TN}{TP+TN+FP+FN}$

In this case the accuracy is $0.68$. Can I say that I have low accuracy because the value false positive is high? There is any relathion between false positive and the parameters true positive or true negative?

asked 2022-04-06

What is the rationale behind ROC curves?

I am not sure how ROC curves work. I see that the X-Axis is the false positive rate while the Y axis is the true positive rate.

1) I don't understand how for a given statistical learning model, you could have the true positive and false positive rate to vary from 0 to

1. Are you changing parameters in the model to make it so?

2) What about true negatives and false negatives? How are these represented in the curve?

I am not sure how ROC curves work. I see that the X-Axis is the false positive rate while the Y axis is the true positive rate.

1) I don't understand how for a given statistical learning model, you could have the true positive and false positive rate to vary from 0 to

1. Are you changing parameters in the model to make it so?

2) What about true negatives and false negatives? How are these represented in the curve?

asked 2022-05-10

"A certain disease has an incidence rate of 2%. If the false negative rate is 10% and the false positive rate is 1%, compute the probability that a person who tests positive actually has the disease."

Why do we need to use Bayes' Theorem for this question?

Why do we need to use Bayes' Theorem for this question?

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A pregnancy test kit is 98.5% accurate for true positive result, i.e. the result is positive when the tester is actually pregnant. If she is not pregnant, however, it may yield a 0.8% false positive. Suppose a woman using this pregnancy kit is 60% at risk of being pregnant.

Not sure about her first test which turned out to be negative, the woman decides to take the test again. This second test, however, turns out to be positive. Assuming the two test are independent, find the probability that she is actually pregnant.

Now she is so confused whether or not she is pregnant. So she take the tests n more times and the results for these n more tests are all positive. Find the minimum value for n so that she can be at least 99.99% sure of pregnancy, assuming all test are independent.

Not sure about her first test which turned out to be negative, the woman decides to take the test again. This second test, however, turns out to be positive. Assuming the two test are independent, find the probability that she is actually pregnant.

Now she is so confused whether or not she is pregnant. So she take the tests n more times and the results for these n more tests are all positive. Find the minimum value for n so that she can be at least 99.99% sure of pregnancy, assuming all test are independent.

asked 2022-04-30

3% of the population has disease X.

A laboratory blood test has

(a) 96% effective at detecting disease X, given that the person actually has it.

(b) 1% “false positive” rate. i.e, a person who does not have disease X has a probability of 0.01 of obtaining a test result implying they have the disease.

What is the probability a person has the disease given that the test result is positive?

A laboratory blood test has

(a) 96% effective at detecting disease X, given that the person actually has it.

(b) 1% “false positive” rate. i.e, a person who does not have disease X has a probability of 0.01 of obtaining a test result implying they have the disease.

What is the probability a person has the disease given that the test result is positive?