Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

SIT1001 Probability and Statistics I

Tutorial 4

1. An urn contains 7 red and 11 white balls. Draw one ball at random from the  urn. Let ��� = 1 if a red ball is drawn, and let ��� = 0  if a white ball is drawn.  Give the pmf, mean and variance of ���.

[7/18,77/324]

2. According to a CNN/USA Today poll, approximately 70% of Americans  believe the IRS abuses its power. Let ��� equal the number of people who believe the IRS abuses its power in a random sample of ��� = 25 Americans. Assuming that the poll results are still valid, find the probability that

a. ��� is at least 13.

b. ��� is at most 11.

c. ��� is equal to 12.

d. 

Give the mean, variance and standard deviation of ���.

[0.9825;0.0060;0.0115;17.5,5.25,√5.25]

3. Let ��� equal the proportion of all college and university students who would say yes to the question, “Would you drink from the same glass as your friend if you suspected that this friend were an AIDS virus carrier?” Assume that ��� = 0.10. Let ��� equal the number of students out of a random sample of size ��� = 9 who would say yes to this question.

a. How is ��� distributed?

b. Give the values of the mean, variance and standard deviation of ���.

c. Find (i) ���(��� = 2), and (ii) ���(��� ≥ 2).

[���(9,0.10);0.9,0.81,0.9;0.1722,0.2252]

4. A random variable ��� has a binomial distribution with mean 6 and variance 3.6. Find ���(��� = 4).

[0.1268]

5. It is claimed that for a particular lottery, 1/10 of the 50,000,000 tickets will win a prize. What is the probability of winning at least one prize if you purchase (a) 10 tickets or (b) 15 tickets?

[0.6513,0.7941]

6. Define the pmf and give the values of ���, ���² and ���  when the moment generating

8. A CD player has a magazine that holds six CDs. The machine is capable of randomly selecting a CD at random and then selecting a song randomly from that CD. Suppose that five CDs are albums by Paul McCartney and one is by Billy Joel. The player selects songs until a song by Billy Joel is played after which the machine is turned off. Give the probability that the machine is turned off after

a. The sixth song.

b. At least five songs have been played.

c. Suppose now that the songs continue to be played until the second song by Billy Joel has been played. Find the probability that at most four songs are played.

[0.0670,0.4823,0.1319 ]

9. One of four different prizes was randomly put into each box of a cereal. If a family decided to buy this cereal until it obtained at least one of each of the four different prizes, what is the expected number of boxes of cereal that must be purchased?

[25/3]

10. Customers arrive at a travel agency at a mean rate of 11 per hour. Assuming  that the number of arrivals per hour has a Poisson distribution, give the probability that more than 10 customers arrive in a given hour.

[0.540]

11. Flaws in a certain type of drapery material appear on the average of one in 150 square feet. If we assume the Poisson distribution, find the probability of at most one flaw in 225 square feet.

[0.558]

12. A roll of biased die results in a two only 1/10 of the time. Let ��� denote the number of twos in 100 rolls of this die. Approximate

a.   ���(3 ≤ ���  ≤ 7). b. ���(��� ≥ 5).

[0.217,0.971]

13. Let ��� equal the number of telephone calls per hour that are received by 911 between midnight and noon and reported in the Holland Sentinel. On October 29 and October 30, the following numbers of calls were reported: 0,1,1,1,0,1,2,1,4,1,2,3,0,3,0,1,0,1,1,2,3,0,2,2

a. Calculate the sample mean and sample variance for these data. Are they approximately equal to each other?

b. Assume that ��� = 1.3. Draw a probability histogram for the Poisson distribution and a relative histogram of the data on the same graph.

c. Does it look like Poisson distribution with ��� = 1.3 could be a reasonable probability model based on these limited data?

[4/3,88/69,yes;yes]

14.  A store selling newspapers only orders ��� = 4 of a certain newspaper because the manager does not get many calls for that publication. If the number of requests per day follows a Poisson distribution with mean three,

a. What is the expected value of the number sold?

b. How many should the manager order so that the chance of running out is less than 0.05?

[1.941,6]

15. Solve Question 13 using R.