Stat 1880 worksheet problems

Question

Question 1. If 𝑋 represents the number of heads obtained in three tosses of a fair coin, (Create the table for both parts)

(a) Find the probability distribution of 𝑋.

(b) Find the expected value and the standard deviation.

Question 2. A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 20% of students attend Tet festivities. We are interested in the number of students who will attend the festivities.

1. (a) Give the distribution of 𝑋. 𝑋 ~ _____(_____,_____)
2. (b) How many of the 12 students do we expect to attend the festivities?
3. (c) Find the probability that at most four students will attend.
4. (d) Find the probability that more than two students will attend.
5. (e) Findthestandarddeviation.

Question 3. Suppose that about 80% of graduating students attend their graduation. A group of 25 graduating students is randomly chosen.

1. (a) Give the distribution of 𝑋. 𝑋 ~ _____(_____,_____)
2. (b) How many are expected to attend their graduation?
3. (c) Find the probability that 18 students attend.
4. (d) Find the standard deviation.

Answer 1

Note: I am only addressing question 1 and I will be using R Markdown for some things.

1a. Because we can talk about X as the number of successes, it is appropriate to use the binomial or the beta distributions. I will use the binomial for this problem. For the binomial, we have x = number of successes, n = number of trials, and p = probability of success. I am assumming that n=3 in this problem, since no other information is given and if it is a fair coin, the probability of getting heads would be 0.5, since we only have 2 sides of a coin. Thus, our pmf can be defined as the following:

$p_x=\binom{3}{x}(0.5)^x(1-0.5)^{3-x}$ #for x={0,1,2,3}

1b. The expected value and standard deviation of the binomial distribution are known to be np and √(np(1-p)), respectively. Thus, the expected value is 3*0.5=1.5 and the standard deviation is √(1.5(0.5))≈0.866