# Central Limit Theory and the Binomial Distribution Research Paper

**Pages:** 5 (1285 words) ·
**Bibliography Sources:**
3 · **Level:** Master's · **Topic:** Mathematics

SAMPLE EXCERPT:

[. . .]
6%.

c) When you invest in a large number of stocks chosen at random, your average return will approach a Normal distribution.

The correct statement is that if you invest in more and more stocks selected at random, then your average return on these stocks will get closer and closer to 10.6%. This is because the law of large numbers states that of the size of the sample is large, then there is the tendency that the mean will converge to a number, which is very close to the population mean.

Part 2

1. Ray Allen, who is one of the best 3-point shooters over the last 15 years in the NBA, has a chance to shoot four free throws. He was fouled shooting a 3-point shot, and he gets to take a fourth shot due to a technical foul. Suppose that the probability that he makes a free throw is .9, and his free throws are independent of each other. Let X be the random variable that gives the number of free throws made in 4 attempts.

a) Give the possible values for X. What type of distribution model is appropriate? Explain why.

The possible values for X are 0, 1, 2, 3, and 4

b) Obtain the probability distribution for x

c) Indicate what the probability is for Ray Allen to make three or more of the free throws

P ( X ? 3) =

P (0) = [4! / (0! (4 - 0)!] 0.90 (1 – 0.9)4-0

= [(4! / 0! 4!)] 1(0.1)4

=0.0001

P (1) = [4! / (1! (4 - 1)!] 0.91 (1 – 0.1)4-1

= [4! / (1! (3)!] 0.9 (0.1)3

= 0.0036

P (2) = [4! / (1! (4 - 1)!] 0.92 (1 – 0.1)4-2

= [4! / (1! (3)!] 0.92 (0.1)2

= 0.0324

Probability = 0.0001 + 0.0036 + 0.0324 = 0.0361

P ( X ? 3) = 1 – 0.0361

= 0.9639

2. Suppose that in a quality control study it was determined that for a perishable item the number X of defective items follows a binomial distribution with n= 20 and p = .10.

a) Find the probability that a sample of 20 contain no more than two defective items.

b) If one were to take many samples of 20 and found the mean number of defectives in each sample, what should we expect for the mean and standard deviation of the number of defectives?

3. This problem is motivated by the study: http://site.iugaza.edu.ps/wdaya/files/2013/03/A-Random-Walk-Down-Wall-Street.pdf

The random walk theory of stock markets implies that an index of stock prices has probability 0.63 of increasing in any year. In addition the change in the index in any year is not influenced by whether it goes up or down in an earlier year. Suppose that X is the number of years among the next six years in which the index rises.

a) Assuming that X has a binomial distribution, give are the values of n and p.

N = 6 and P = 0.63

b) What is the set of values that X can attain?

X can be either 1, 2, 3, 4, 5 or 6

c) Calculate the probability of each of the values of X given in part b. You can use the Excel BINOM.DIST function. Include the Excel function used and its result in your Word document.

P (x = 1)

=BINOM.DIST(1,6,0.63, FALSE)

= 0.02621

P (x =2)

BINOM.DIST(2,6,0.63, FALSE)

= 0.1116

P (X = 3)

= BINOM.DIST(3,6,0.63, FALSE)

= 0.2533

P (X = 4)

= BINOM.DIST(4,6,0.63, FALSE)

= 0.3235

P (X= 5)

= BINOM.DIST(5,6,0.63, FALSE)

= 0.2203

P (X = 6)

= BINOM.DIST(6,6,0.63, FALSE)

= 0.0625

d) Use Excel to construct a bar chart for the probabilities. The values of X should be on the horizontal axis, and the probabilities on the vertical axis. For information on how to use…
[END OF PREVIEW]

### Four Different Ordering Options:

### Compare the Four Ordering Options

- 1.Buy this paper with your credit card or cash balance at PayPal. Within 10 hours, we'll send the Microsoft Word file to the email address on your PayPal account.
- 2.Same as #1, but we will also remove the paper from our site for 30 days!
- 3.Need this paper immediately? Want to individually download any of our 175,000+ exclusive, private, non-plagiarized papers for 30 days? It takes only 2 minutes to subscribe and get instant access!
- 4.One of our highly experienced experts will write a brand new, 100% unique paper matching the exact specifications and topic that you provide! You'll be the only person on the planet to receive the one-of-a-kind paper that we write for you! Use code "Save10" to save 10% on your 1st order!

1. Buy the full, 5-page paper: $24.68

or

2. Buy & remove for 30 days: $38.47or

3. Monthly access to 175,000 papersor

4. Let us write a NEW paper for you!

Most popular!

#### Normal Distribution Central Limit Theorem and Point Estimate and an Interval Term Paper …

#### Theory Building Term Paper …

#### Theory Borrowing in the Academic Disciplines of Communication and Women's Studies Research Paper …

#### Theory -- Horotwitz & Bartholomew Attachment Chapter Writing …

#### Theory Essay …

Cite This Research Paper:

APA Format

Central Limit Theory and the Binomial Distribution. (2018, March 16). Retrieved January 19, 2019, from https://www.essaytown.com/subjects/paper/central-limit-theory-binomial-distribution/9388496MLA Format

"Central Limit Theory and the Binomial Distribution." 16 March 2018. Web. 19 January 2019. <https://www.essaytown.com/subjects/paper/central-limit-theory-binomial-distribution/9388496>.Chicago Format

"Central Limit Theory and the Binomial Distribution." Essaytown.com. March 16, 2018. Accessed January 19, 2019.https://www.essaytown.com/subjects/paper/central-limit-theory-binomial-distribution/9388496.