Mathematics

.1. Let X and Y be random variables taking the values on the set 0 ? X ? Y with joint density functionfX,Y (x, y) = (4e?3xe?yfor 0 ? x ? y0 otherwise.(a) Draw the support of the joint density function. Are X and Y independent? Why or why not?(b) Find the marginal density function for Y .(c) Write a formula for the conditional density of X given that Y = y for an arbitrary value of y,fX|Y (x|y).(d) Compute the conditional expectation E(X|Y = y). The result should depend on the value of ybut not x.(e) Based on your answer to part (d), do you think the covariance of X and Y should be positive,negative, or zero? Explain.2. The number of emails I will receive as I sleep tonight is a Poisson random variable with mean 30. Eachtime I receive an email, the probability that it is spam is 0.3, independent of all other emails. Thusif I receive N emails, the number of emails which are spam is a Bin(N, 0.3) random variable. Let Nbe the number of emails I will receive tonight, and let X be the number of spam emails I will receivetonight.(a) Use “double expectation” to compute the expected value of the number of spam emails I willreceive tonight, E(X).(b) Use “double expectation” to compute the variance of the number of spam emails I will receivetonight, Var(X). You may need to use the fact that E(N2) = E(N)2+Var(N).(c) Without doing any computations, do you expect the covariance of X and N to be positive,negative, or zero? Explain why.(d) Use “double expectation” and your answer to part (a) to compute the covariance of N and X.Hint: The covariance is E(XN)?E(X)E(N). Since E(N) = 30 and you found E(X) in part (a),you just need to find E(XN). Conditioning on N, this is E(E(XN|N)) = E(NE(X|N)), sincewhen conditioning on N, we are thinking of N as a constant. You will have to use the fact thatE(N2) = E(N)2+Var(N).(e) Use your answers to parts (b) and (d) to find the correlation coefficient between X and N.. A recent NYT/Sienna poll showed that in the upcoming presidential election, 36% of Americans planto support President Trump, 50% plan to support the presumptive Democrat nominee, Joe Biden,and the remaining 14% are undecided. Assume these numbers are correct, and suppose we gather5 randomly selected Americans in a room. Find the probability that there are the same number ofTrump supporters as Biden supporters in the room. That is, if T is the number of Trump supporters,and B is the number of Biden supporters, find P(T = B).1

1. Build a frequency distribution table with frequencies (count), relative frequency and cumulative frequency. 2. Using Excel create the frequency table and insert a bar graph using the class interval levels and the frequencies from each class interval. 3. Create a pie chart using the relative frequencies. 4. Calculate the range 5. Calculate the mean, median, the mode 6. Calculate the sample standard deviation and variance. 7. Determine any outliers 8. Determine which is the best way to determine the average score using the mean, the median or the mode. Explain why? 9. Determine the 5 number summary. 10. Calculate the interquartile range (IQR)

 Reflect on your professional or personal world. What is an example of a situation thatyou might be able to use an equation with a single unknown to help understand? What isan example of a situation that you might not be able to use an equation with a single unknown to understand? What makes an equation with a single unknown helpful in one ofyour examples but not the other? What patterns exist in your two examples that might be helpful in determining when to use a simple equation? 

Calculus, originally called infinitesimal calculus or “the calculus of infinitesimals”, is the mathematical study of continuous change. When they were introduced who objected to them.

Imagine that a friend is starting college and wondering whether to take calculus. The friend asks you “What does calculus have to do with the real world?” Answer the question and cite all sources that you use, including the textbook.

Note: For all the questions, provide detailed solution steps. Question 1: (80 points, 20 each) a. Find the equation of the line that passes through the point (-6, 3) and has the slope 7.  b. Find the equation of the line that passes through the points (4, -3) and (12, 1).  c. Find the equation of the line with an x-intercept of 3 and a y-intercept of -4. d. Find the equation of the line if y-intercept is 4, and the line is parallel with the x-axis.  Question 2:   (10 points, 5 each) Graph the function: a) 2x -3y = -4. b) 4 + 2y = 3x. Question 3: (Bonus).  (10 points/5 each) The budget of a company per month is given by the equation F(x) = 4x-2, find the budget: a. After 5 years. b. After 7.5 years.

Chapter 6: More Practice With Normal Distributions. As a part of preparation for this week’s test, let’s practice with another case related to the Normal distribution. Assign your numbers for mean ? and standard deviation ?. Then select a number “A” below mean ?, and a number “B” above mean ?. Use Appendix Table for the Normal Distribution to find the probability that x is between A and B: P (A < x < B). Here are steps to follow: convert A to z score (let's call it za), convert B to z score (let's call it zb).; From Appendix table find area under curve to the left of za and to the left of zb. That will give you P (z < za) and P (z < zb). If za or zb are not in the table, change your A or B. Use formula: P (A < x < B) = P (za < z < zb) = P (z < zb) - P (z < za) Don't just assign numbers, make an example from a real situation where this technique can be applied. For instance, see Chapter 6 section 6.3 example 6.11 on page 281 of eText: Intelligence Quotients example. Create your own scenario and values.

Using the Internet, the text, or another reliable source such as a newspaper or periodical site, (not a scholastic or school site like Khan Academy and not Wikipedia), research some important data that has been presented in graph or table format. Make sure to cut and paste the chart or table; if that is not possible, paste a link to the webpage. (Check it to be sure it links correctly.) Present your graph and fully explain the information it is representing. Pose two questions regarding the data and graph or chart for your fellow students to consider and respond to in your post.

Looking for someone to answer my questions and pretty much be at my side for the exam. If I do not know an answer I will send you the problem and you must give step- by – step instructions to answer the question. 4 minutes per question are given. Total of 2 hour exam. Needs to be started by 10pm.

Given the following polynomial function: