Question

Write solutions legibly, and show all work. Walk the reader through your thought process, using English...

Write solutions legibly, and show all work. Walk the reader through your thought process, using English words when necessary.

1. Recall question 2 of the previous homework – We draw 6 cards from a 52 card deck and let X = the number of heart cards drawn. You already found the pmf back then. You’re allowed to use it here without re-deriving it. a. What is the expected value of X? b. What is the variance of X? What is the standard deviation of X?

2. A man has seven keys on a key ring, one of which fits the door he wants to unlock. He randomly selects a key and tries it. If it doesn’t work, he selects a different key at random and tries that one (so he’s sampling without replacement). He continues in this manner until the door opens. Let X = the number of keys he tries until opening the door (counting the key that actually works). Find the expected value and standard deviation of X.

3. Suppose that the average number of equipment breakdowns at a factory per week is 5. Later we’ll show that the average is our best guess at the expected value of ? = ?ℎ? ?????? ?? ????????? ?????????? ?? ? ????? ???? (assuming that this distribution stays the same across multiple weeks.) It’s also known that, through some other sample statistic, that our best guess at the standard deviation of X given the data is 0.8. a. With the help of our boy Chebyshev, find an interval such that we can be at least 90% that the number of breakdowns next week will fall within that range. b. Suppose that the supervisor promises the board of directors that the number of breakdowns will rarely exceed 8 in a one- week period. Is the supervisor safe in making this claim? Why or why not? c. Google Chebyshev and just stare at this guy’s magnificent beard for a few seconds. You don’t need to write anything down. You’re welcome.

4. Let ? be some random variable with mean ? and standard deviation ?. Define ? = ?−? ? . What is the mean of ?? What is the standard deviation of ?? (This general fact about random variables will come in handy later when we get to the normal distribution.)

5. Suppose we toss a fair coin twice. Let X = the number of heads, and Y = the number of tails. X and Y are clearly not independent. a. Show that X and Y are not independent. (Hint: Consider the events “X=2” and “Y=2”)
b. Show that E(XY) is not equal to E(X)E(Y). (You’ll need to derive the pmf for XY in order to calculate E(XY). Write down the sample space! Think about what the support of XY is and find those probabilities.)

6. Let ?~???????[0,?], let X = ???(?), and Y = ???(?) a. First, we’ll show that X and Y are not independent (which should be obvious but we still need to be able to show it.) i. What is ?(√2 2 ≤ Y ≤ 1)? (Hint: Put this probability in terms of U) ii. What is ?(√2 2 ≤ ? ≤ 1 | √2 2 ≤ X ≤ 1)? (Hint: ↑) b. What we’ve shown so far is that there is a clear causal relationship between X and Y. Next we’ll show that, despite X and Y being very, very dependent, they are still nonetheless uncorrelated. i. Find E(X), E(Y), and then multiply them together to get E(X)E(Y). (Hint: Use the Law of the Unconscious Statistician) ii. Find E(XY). (Hint: ↑) iii. The correlation between Y and Z is the square root of E(XY)-E(X)E(Y). What is it?
(If you’re interested in what this means in terms of the larger correlation vs causation argument, see the note at the end of the assignment)

7. Suppose that the mean exam score for a college class is a 60 (where the highest possible score is 100), and the standard deviation is 10. If there are 50 people in the class, what is the maximum possible number of A’s that there could have been? (Assume that getting an A means getting a score greater than or equal to 90) (Hint: What does Chebyshev’s Inequality tell you about scores between 30 and 90? That is, scores which are within 3 standard deviations of the mean?))

1:

Here X has hypergeometric distribution with following parameters:

Population size: M = 13

Number of successes in population, that is number of hearts: k=12

Sample size: n=6

(a)

The mean is

(b)

The variance is

The standard deviation is

Earn Coins

Coins can be redeemed for fabulous gifts.

Similar Homework Help Questions
• Please write neatly and legibly. Please show all work. 1. Recall that given a basis, the...

Please write neatly and legibly. Please show all work. 1. Recall that given a basis, the space of linear endomorphisms of R", End (R"), can be identified with the space of nxn matrices. Let us denote this space by Mat (n). Clearly, with respect to standard addition of matrices and multiplication by scalars, Mat (n) is a na-dimensional vector space. 1. Let X e Mat (n). Then, we can think as being coordinates on Mat (n). 1,j=1...n Clearly, we must...

• I want all parts answers, and the answer needs to correct. the answers need include from...

I want all parts answers, and the answer needs to correct. the answers need include from part a) to part n) h.) E(X) (Give decimal answer to two places past the decimal.) Tries 0/5 1.) E(Y) (Give decimal answer to two places past the decimal.) ries 0/5 j.) E(XY) (Give decimal answer to two places past the decimal.) Tries 0/5 k.) Cov(X,Y) (Give decimal answer to THREE places past the decimal.) Tries 0/5 1.) Standard deviation of X (Give decimal...

• Please show work so that I can see the process also Let’s say you scored a 111 on exam 1 and you scored an 125 on exam 2. You can predict your final exam score with the following prediction equation:...

Please show work so that I can see the process also Let’s say you scored a 111 on exam 1 and you scored an 125 on exam 2. You can predict your final exam score with the following prediction equation: Y’ = bX + c (round to nearest whole number). X is the total number of points you earned on the first two tests. Given: Mean = 120; standard deviation of y = 100. The correlation (r) between the total...

• Please answer all rest questions. Thanks! A service station has both self-service and full-service islands. On...

Please answer all rest questions. Thanks! A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation. P(x,y) у 0 1 2 0...

• There are two traffic lights on a commuter's route to and from work. Let X, be...

There are two traffic lights on a commuter's route to and from work. Let X, be the number of lights at which the commuter must stop on his way to work, and X, be the number of lights at which he must stop when returning from work. Suppose that these two variables are independent, each with the pmf given in the accompanying table (so X,, X, is a random sample of size n 2) 1 2 1.5, -0.65 0.2 0.1...

• please show your work, especially for question one! thanks Question 1. Here are some data: 115.8,...

please show your work, especially for question one! thanks Question 1. Here are some data: 115.8, 115.2, 114.6,115.9, 116.4 Use the table below to answer the following questions (NOT R!): a) Compute the mean value for these data b) Compute the deviation from the sample mean for each observation (xi-x) cCompute the sample standard deviation using the defining formula d) Compute the sample standard deviation using the computational short-cut e) What is the mean value of the deviations? xi-x)2 xi...

• This is Probability and Statistics in Engineering and Science Please show your work! especially for part B A Poisson distribution with λ=2 X~Pois(2) A binomial distribution with n=10 and π=0.45. X~bin...

This is Probability and Statistics in Engineering and Science Please show your work! especially for part B A Poisson distribution with λ=2 X~Pois(2) A binomial distribution with n=10 and π=0.45. X~binom(10,0.45) Question 4. An inequality developed by Russian mathematician Chebyshev gives the minimum percentage of values in ANY sample that can be found within some number (k21) standard deviations from the mean. Let P be the percentage of values within k standard deviations of the mean value. Chebyshev's inequality states...

• Please show how did you came up with the answer, show formulas and work. Also, please do Parts e to i. Thank you so much 1. Consider the following probability mass function for the discrete joint pro...

Please show how did you came up with the answer, show formulas and work. Also, please do Parts e to i. Thank you so much 1. Consider the following probability mass function for the discrete joint probability distribution for random variables X and Y where the possible values for X are 0, 1, 2, and 3; and the possible values for Y are 0, 1, 2, 3, and 4. p(x,y) <0 3 0 4 0.01 0 0 0.10 0.05 0.15...

• Please show all work X, be a random sample from the distribution with the probability density...

Please show all work X, be a random sample from the distribution with the probability density function Let A0 and let X, X2, f(x; A) 24xe, x>0. a. Find E(X), where k> -8. Enter a formula below. Use* for multiplication, for divison, ^ for power, lam for A, Gamma for the r function, and pi for the mathematical constant . For example, lam k*Gamma(k/2)/pi means Akr(k/2)/T Ax2 or u =x2. Hint 1: Consider u -e"du Hint 2: I'(a) a 0...

• show your work please 7.) The cost of homes in the East Los Angeles area is...

show your work please 7.) The cost of homes in the East Los Angeles area is found to be normally distributed witha mean of \$350,000 with a standard deviation of \$75,000? What can be said about homes that cost \$200,000 or more in the greater Los Angeles Area? GC-4 8.) Notation. Do not state the name. State what it represents in Statistics b. s a. N: c. d. n e. f 9) Use z-score to answer the following a. In...