(REALLY NEED HELP WITH QUESTIONS 1-11 WITH FULL EXPLANATIONS. HAND WRITTEN ANSWERS ARE PREFERRED IF AND WHEN POSSIBLE). (NOTE: Week6-1.txt, Week6-2.txt, and Week6-3.txt data is listed at the bottom at the end of the questions)

PLEASE HELP!!!

Recall some definitions and notation that might be useful for these problems. Discrete Case: Let f(x) denote the probability mass function (pmf) of a random variable X of the discrete type with sample space S. The expected value of x or the mean of X is given by

• =E[X]=xSxf(x).

In general, for a function g(x), we have

• E[g(X)]=xSg(x)f(x)

The variance of X is given by

• 2=Var(X)=E[(X)2]=xS(x)2f(x)=E[X2]2

and the standard deviation of X is

• =Var(X)

when Var(X) is positive.

Continuous Case: Let f(x) denote the probability density function (pdf) of a random variable when Var(X) is positive X. The expected value ofX or mean of X is:

• =E[X]=xf(x)d(x)

The variance of X is given by

• 2=Var(X)=E[(X)2]=(x)2f(x)dx

The standard deviation of X is

• =Var(X)

1. (a) Assume that X is a uniform distribution of the first 10 positive integers. Using the definition of the mean above, find the mean of X.

(b) Using the same distribution as part (a), find the standard deviation of X, again using the definition.

2. Let X be a random variable with probability mass function equal to

• f(x)=9(x+1)2

for x = 1, 0, 1. Find:

• (a) E[X]
• (b) E[X²]
• (c) E[3X² 2X + 4]

3. Let and ² denote the mean and variance of a discrete random variable X. Determine

• E[X]andE[(X)2]

which are the mean and variance of the discrete random variableX

4. According to a national study, approximately 70% of Americans believe that the IRS abuses its power. Let X equal the number of people who believe the IRS abuses its power in a random sample of n = 25 Americans. Assuming the study results are still valid, find the probability that

• (a) X is at least 13.
• (b) X is at most 11.
• (c) X is equal to 12.
• (d) Find the mean, variance, and standard deviation of X.

[Hint: Since there are two choices an American can pick (i.e. one believes the IRS is abusing its power, or not), this is a binomial distribution.]

5. Assume there is a collection of 40 bags of tulips, and each bag contains a total of 25 red and yellow tulips together. Of the 40 bags, 30 of them contain 5 red and 20 yellow tulips. The other 10 bags contain 15 red and 10 yellow tulips. A bag is selected at random and one bulb is pulled form the bag. Find the following probabilities:

• (a) a red tulip is chosen
• (b) a yellow tulip is chosen
• (c) If the tulip chosen is red, find the conditional probability it came from one of the bags that has 15 red and 10 yellow bulbs

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

7. In this problem, refer to the data in Week6-1.txt. The data in this file represents the number of visitors to a major website over 300 one-second intervals. You will determine if the observations suggest that they are coming from a Poisson distribution with mean = 3 by answering the questions below. You are encouraged to use Python on this problem.

• (a) Find the frequencies of each value.
• (b) Calculate the sample mean and sample variance. Are they approximately equal to each other?
• (c) Construct a probability histogram with =3 and a relative histogram on the same graph.
• (d) Based on your answers to the parts above, do you think the data comes from a Poisson distribution with mean = 3? Why or why not?

8. Let the random variable X have the pdf f(x) = 2(1x) for 0 x 1, and zero elsewhere.

(a) Sketch the graph of this pdf.

(b) Calculate the distribution function F(x) of f(x) and sketch F. Recall

F(x)=P(Xx)=f(t)dt

(c) Find

• i. P(0 X 1/2)
• ii. P(1/4 X 3/4)
• iii. P(X = 3/4)
• iv. P(X 3/4)

(d) Find , , and ² for this distribution.

9. In this problem, refer to the data in Week6-2.txt. One hundred observations of X having a certain distribution were taken, ordered, and noted in the file. You are encouraged to use Python on this problem.

• (a) Calculate the sample meanx
• (b) Calculate the sample standard deviation s.
• (c) Group these data into 10 classes: [0, 0.200], (0.200, 0.400], . . . , (1.800, 2.000]. Con- struct a relative frequency histogram using these intervals.
• (d) Now superimpose on the graph from the previous part the pdf f(x) = x/2 for 0x2. Is this pdf a good fit for the data? Why or why not?

10. Let X have an exponential distribution with mean = 20.

(a) Compute

• i. P(10 < X< 30)
• ii. P(X > 30)
• iii. P(X > 40|X > 10)

(b) In this problem, refer to the data in Week6-3.txt. You are encouraged to use Python on this problem. The data in the file was simulated from an exponential distribution with = 20. Compare the relative frequencies of the appropriate events in the data with the respective probabilities found in the parts above.

11. If X is ²(23), find the following:

• (a) P (14.85 < X < 32.01)
• (b) The mean and the variance of X

Week6-1.txt 5 5 9 3 3 4 8 3 8 5 1 10 5 3 6 7 4 5 4 3 2 1 5 6 5 4 2 1 1 4 4 4 7 3 5 2 2 6 10 4 3 9 9 5 2 5 8 0 3 4 3 5 5 5 5 3 7 5 0 4 1 4 7 5 5 4 3 1 7 5 4 4 4 9 2 4 3 3 5 3 6 3 3 8 4 3 4 10 3 4 5 5 6 3 4 6 8 3 6 4 4 9 2 3 3 7 8 4 6 3 2 7 9 9 5 0 5 6 4 4 4 5 6 3 5 5 7 5 3 7 5 4 2 9 5 7 9 8 3 3 2 4 7 10 5 4 4 4 6 4 4 5 6 6 3 5 0 10 8 7 9 3 7 4 2 10 7 8 8 8 9 4 5 9 6 11 3 1 3 3 3 6 5 5 4 3 9 7 7 6 4 4 6 4 6 4 2 3 5 6 4 6 5 8 6 4 4 7 2 3 3 7 6 2 5 5 5 5 5 4 2 5 8 5 6 5 7 4 3 2 4 4 4 4 1 5 4 4 0 2 3 5 7 3 7 4 3 14 3 6 5 9 6 6 5 3 8 7 9 7 2 4 5 4 4 2 2 7 7 6 5 0 7 2 8 0 10 4 3 1 8 2 8 5 7 6 4 1 6 8 3 4 5 5 4 6 6 5 2 9

Week6-2.txt

0.031 0.064 0.07 0.185 0.186 0.193 0.334 0.369 0.373 0.414 0.428 0.43 0.463 0.481 0.498 0.54 0.556 0.562 0.574 0.584 0.588 0.59 0.627 0.627 0.634 0.645 0.681 0.686 0.723 0.731 0.74 0.778 0.793 0.795 0.801 0.829 0.833 0.833 0.851 0.856 0.879 0.914 0.927 0.933 0.944 0.958 0.98 0.997 1.006 1.008 1.05 1.067 1.089 1.105 1.117 1.12 1.149 1.154 1.161 1.166 1.184 1.197 1.203 1.205 1.208 1.217 1.235 1.241 1.268 1.278 1.281 1.285 1.292 1.327 1.347 1.376 1.382 1.426 1.451 1.478 1.503 1.503 1.505 1.556 1.566 1.572 1.587 1.589 1.638 1.643 1.65 1.703 1.748 1.794 1.795 1.817 1.834 1.908 1.994 1.997 Week6-3.txt

34.368 20.159 43.14 14.032 7.731 53.489 0.519 10.679 9.059 50.13 7.517 4.907 40.161 9.205 7.844 1.465 6.156 6.351 5.312 22.878 12.075 12.882 0.413 9.564 16.751 17.086 1.817 53.177 8.995 18.315 21.632 14.366 8.81 2.766 9.539 1.564 7.042 16.245 84.691 94.82 10.819 55.754 32.733 16.545 6.598 15.682 1.903 3.345 5.887 3.618 7.606 5.151 29.028 3.914 23.901 1.227 1.988 0.156 13.584 28.215 6.376 118.678 19.196 8.722 25.985 18.786 12.769 3.507 8.781 54.717 17.463 10.435 56.044 7.292 24.246 13.824 84.179 37.692 5.984 25.524 39.354 17.41 1.893 16.537 6.454 44.035 0.344 1.83 4.478 2.151 3.858 25.826 1.412 14.986 25.132 77.812 2.172 28.185 21.131 21.603