Practice Math Statistics Problems and Master the Concepts

What sort of qualitative behaviour does a stock following a process of the form
$d S_{t} = α (μ - S_{t}) d t + S_{t} σ d W_{t}$
exhibit? What qualitative effects do altering $μ$ and $α$ have? What effects do they have on the price of a call option?
My try:
Assume that $α < 0,$ . If $S_{t} < μ,$ , then $α (μ - S_{t}) > 0.$ . Similarly, if $S_{t} > μ,$ , then $α (μ - S_{t}) < 0.$ . This means that if $α < 0,$ , then $S_{t}$ is mean-reverting.
Conversely, if $α < 0,$ , similar analysis above implies that $S_{t}$ is trend following.
In the derivation of Black-Scholes call option price, mean rate of a geometric Brownian motion does not affect the price as it will be changed into risk-free rate.
As the volatility is the same, $α$ and $μ$ do not have any effect on the price of a call option.
Am I correct?

Quarrippesspevmo 2022-06-04

How do you find the mean, median and mode of x, x, x, x, x+y, x+y, x+2y, x+2y, x+2y, x+2y?

or5a2dosz80z 2022-06-04

Significance level for a hypothesis test for a linear regression
Consider linear regression model $Y_{i} = a + b \cdot x_{i} + ϵ_{i}$ , $i = 1, 2, 3, 4, 5$ where $a, b \in R$ are unknown and $x_{1} = x_{2} = 1, x_{3} = 3, x_{4} = x_{5} = 5$ , $ϵ_{i}$ are iid, normally distributed with mean =0 and variance =9. Consider the hypothesis $H_{0} : b = 0$ with the alternative $H_{1} : b \neq 0$ , with the critical region ${| \hat{b} | > c}$ , where b^ is a maximum likelihood estimator and c is chosen in such a way that the significance level of a test is equal to 0,05. Calculate c.
Could you help me with this exercise? It is taken from the actuary exam organized in my country. I thought that I am able solve this exercise, however my answer c=0,7528 is wrong, the correct answer is c=1,47.
Edited: the exercise seems very easy, but I'm sure that my method of solution is wrong, as I've seen the similar exercise and my method results with the wrong answer as well.
That's why I've decided to start a bounty, however I do not know how (I can see "question eligible for bounty in 59 minutes" only, not "start a bounty")

Noah Dominguez 2022-06-04

Does a data set always have a mode?

Sara Bella2022-06-04

An educational psychologist wishes to know the mean number of words a third grader can read per minute. She wants to make an estimate at the $99 %$ level of confidence. For a sample of $1869$ third graders, the mean words per minute read was $25.3$ . Assume a population standard deviation of $5.3$ . Construct the confidence interval for the mean number of words a third grader can read per minute. Round your answers to one decimal place.

Sara Bella2022-06-04

A survey of several 10 $10$ to 11 $11$ year olds recorded the following amounts spent on a trip to the mall:
$23.22,$9.71,$14.34,$23.05,$16.61,$7.22,$22.15

Construct the 99% confidence interval for the average amount spent by $10$ to 11 year olds on a trip to the mall. Assume the population is approximately normal.

Step 4 of 4 :

Construct the 99% confidence interval. Round your answer to two decimal places.

hicaderasu1c8q 2022-06-03

Probability of determinants being coprime
I have a question that is not of particular significance, but I would love to understand the underlying principles.
Suppose we have two square $3 \times 3$ matrices, $M_{1}$ and $M_{2}$ with
$M_{1} = (\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}) and M_{2} = (\begin{matrix} b_{1} & b_{2} & b_{3} \\ b_{4} & b_{5} & b_{6} \\ b_{7} & b_{8} & b_{9} \end{matrix})$
with the coefficients $a_{n}, b_{n} \in Z$ and $1 \leq a_{n}, b_{n} \leq 9$ What is the probability that the matrices' determinants are coprime, when uniformly random coefficients satisfying the conditions are chosen.
I am familiar with the Riemann's $ζ$ function way to find out the probability of two random integers being coprime, but I have no clue how to apply that here with additional conditions on the numbers given.
I did test it mechanically, using Mathematica and the result is around 30%, but I would like to see a proper way to do it.
I would love to at least get a few pointers as what to research to tackle this problem.
Thank you very much!

Quarrippesspevmo 2022-06-03

How do you find the mean of the set of numbers 7, 10, 9, 13, 6?

Warren Leach 2022-06-03

Anita has two sisters and three brothers. The mean of all their ages is 6 years. What will their mean age be 10 years from now? Twenty years from now?

Vidhi Trivedi2022-06-02

From a random sample of 25 dates, the sample mean of record high daily temperatures in Toronto was 25℃ and the standard deviation of this sample is calculated as 3.5℃. Find the 90% confidence interval for the population mean. Explain all your steps.

conIjonnoraj0nls 2022-06-02

The thickness of a non-spreadable liquid between two surfaces
There's this question I've come to when doing my research when working on polymer brushes; what I'm looking for is to find my reaction mixture thickness between two plates. So the setup is as follows: My bottom plate, with initiator on it. Then I put a drop of my reaction mixture on it which does not spread. Lastly, I put the top plate and the drop fills the gap between two plates and forms a film. I know how much the top plate weighs and the surface tension of drop and two plates, how can I calculate the film thickness between two plates? Here's my thought processes:
1.Should I look at the problem unsteady or steady? When thought of steady, I tried to equalize hydrodynamic pressure and weight of top plate but that did not match for the data of a paper working close to my work. They don't mention how they calculated their thickness.
2.Can I think of it simplistically? Like I know the volume of the drop that I put, I know the surface of the plate, then divide the volume by surface to get film thickness?

fulgoripb0b 2022-06-02

Which of the following is not a risk management process?
A. Perform Actuarial Risk Analysis.
B. Perform Qualitative Risk Analysis.
C. Perform Quantitative Risk Analysis.
D. Implement Risk Responses.

Jabari Pitts 2022-06-02

What is the standard deviation of {1, 2, 3, 4, 5}?

Schenone2pare 2022-06-02

A hospital identifies patients with an average false positive rate of 0.01. If it checks 31333 patients and returns a total of 357 positive results, then what's the expected number of true positives?

fwndelet15ukn 2022-06-01

According to my research the Fowlkes-Mallows Index should be a good tool to get a comparable evaluation score, but I am not sure about the correct application.
Currently I calculate the true positives, false positives and false negatives for each cluster, then sum them together for total values and add them in the formula.
$F M = \sqrt{\frac{T P}{T P + F P} \cdot \frac{T P}{T P + F N}}$
where $T P$ is the number of true positives, $F P$ is the number of false positives, and $F N$ is the number of false negatives.
Is this the correct application of the Algorithm?

oplodnjomqunfs 2022-06-01

Determine if the following situations use random or nonrandom sampling. Then, identify what type of random or nonrandom sampling technique is used.
1. Respondents are chosen from a list in the telephone directory.
2. There are 30 freshmen, 20 sophomores, 10 juniors, and 5 seniors enrolled in a certain course. Samples are to be taken from them according to the total number of students per year level.
3. Minority groups of senators are to be interviewed.
4. Every five files out of 500 files will be chosen.
5. Animals going astray are to be observed.
6. Two thousand respondents nationwide, from regions down to barangays, are selected for a national election survey.
7. Five hundred “likes” in Facebook are used as a basis for making a decision.
8. Selected respondents for a study are those with Acquired Immune Deficiency Syndrome (AIDS).
9. Ten names of students were picked out from a box containing 1,000 names written in rolled papers.
10. Selected vehicles are classified according to the last number of their plates as odd or even.

Kale Ware 2022-06-01

Regarding simple random sampling:
a. What is simple random sampling?
b. What is a simple random sample?
c. Identify two forms of simple random sampling and explain the difference between the two.

Shomalit4ob5 2022-06-01

Find the $p$ -value (manually) of the following
Hypothesis testing.
$H_{0} : μ \leq 3 vs H_{1} : μ > 3$
The main thing I know is that
$P (R e j ∣ μ \leq 3) = P (X \geq 3 ∣ μ \leq 3) = e^{- 1} \approx 0.36$
Can I use the $z$ value and use the formula probability of $z$ ?

Thalia Ortiz 2022-06-01

How does the median differ from the mode?

Estrella Le 2022-06-01

Z-Test
$μ < 40 z = - 2.105263158 p = .017634141 \bar{x} = 38 n = 100$
(c) Based on your selection from part (b), what is the value of the standardized test statistic for this significance test?
(d) Based on your selection from part (b), what is the P-value for this significance test?
(e) What is the correct decision for this test, using a 0.05 level of signifcance?
A) Reject the null hypothesis
B) Do not reject the null hypothesis

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