statistics

STA2303 Fall 2014

Assignment 3 (Due Tuesday 10/14/2014)

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Instructions: Your assignment answers should be in complete and grammatically correct sentences. Use this

page as your cover sheet and staple to your assignment.

Problem 1 The following table contains the probability distribution for the number of retransmissions necessary to successfully transmit a 1024K data package through a double-satellite hookup:

X P(X=x)

0 0.34

1 0.35

2 0.25

3 0.05

3 0.01

(a) Define the random variable X. (b) What is its domain? (c) Why is this a discrete probability distribution? (d) Draw a percentage histogram. Don’t forget to label your plot. (e) Compute and interpret the mean of the random variable X. (f) Compute the variance for the random variable X. (g) What is the probability that the necessary number of retransmissions does not exceed 1? Write the

probability in terms of the random variable and compute. (h) What is the probability that at least two retransmission are necessary to successfully transmit the

package? Write the probability in terms of the random variable and compute. (i) From the results above, what conclusions can you reach about the retransmissions?

Problem 2 Three components are randomly sampled, one at a time, from a large lot. As each component is selected, it is tested. If it passes the test, a success (S) occurs; if it fails the test, a failure (F) occurs. Assume that 80% of the components in the lot will succeed in passing the test. Let X represent the number of successes among the three sampled components.

(a) What is the domain of the random variable X? (b) Is X a discrete random variable? Why or why not? (c) Find P(X=3) and provide a statement of interpretation. (d) The event that the first component fails and the next two succeed is denoted by FSS. Find P(FSS). (e) Find P(SFS) and P(SSF). (f) Use the results of parts (c) and (d) to find P(X=2). (g) Write down the probability distribution function for the random variable X. (h) What is this distribution called? (i) Justify that X follows this distribution.

(j) Find P(X=1). (k) Find P(X=0). (l) Find the mean of the random variable X and interpret. (m) Find the standard deviation of the random variable X. (n) Let Y represent the number of successes if four components are sampled. Find P(Y=3). (o) Let Z represent the number of successes if five components are sampled. Find P(Z=3).