# mth 241 fall 2014 8-week homework two.

QUESTION 1.

(a). The random variable X is the number of siblings of a student selected at random from Professor Appiah’s Introductory Statistics class. It’s probability distribution is given in the following table.

 x 0 1 2 3 4 5 P(X=x) 1/4 1/3 1/6 7/48 1/16 1/24

Find the following

(i). The expected value of the random variable X. (3 points)

(ii).The standard deviation of the random variable X. (5 points)

(iii).The variance of the random variable X. (2 points)

(b).Let Y represent the number of Boys in families having three children.

(i). List all possible birth order permutations for families having three children and give the value of Y for each outcome and their associated probabilities. (5 points)

[Hint:  a family ofthree is made up of Boys and Girls]. Use your probability distribution to find the following

(i). The mean of the random variable Y. (3 points)

(ii).The standard deviation of the random variable Y. (5 points)

(iii). The variance of the random variable Y. (2 points)

QUESTION 2.

A die is altered by painting an additional dot on the face that originally had one dot. A pair of such altered dice is rolled.

(a).(i).  Make a table for the sample space of the outcome of this experiment.

(4 points)

(ii).If X is the sum of the number of dots showing on both dice together, what are the possible values of X.(3 points).

(b).(i). Calculate the probabilities ofthe possible values of X from a(ii). (4 points)

(ii).Make a table for the probability distribution of X and verify it is a valid one.

(4 points)

(c).(i). Find E[X(X-1)].  (5 points)

(ii).Find the variance (Var[X]) and the standard deviation (SD[X]). (5 points)

QUESTION 3.

(a).On a multiple choice test of 13 questions, each question has four (4) possible answers, one of which is correct. For students who guess at all answers, find the following if X is the number of correct answers.

(i). The mean of the random variable X. (2 points)

(ii). The standard deviation of the random variable X. (2 points)

(b). Given that the probability function p(x)=(7-x)/12 for x=1,2,3 4 and 5. Find P(1),P(2), P(3), P(4), and P(5). (5 points)

(c).In a city the probability that a person will pass his or her driving test the first attempt is 0.66.Four people are selected at random from among those taking their driving test for the first time. Determine the following.

(i). The probability that exactly one person passed the test.  (3 points)

(ii). The probability that exactly two persons passed the test. (3 points)

(iii) The probability that at least two persons passed the test. (5 points)

(iv). The probability that at least three persons passed the test. (5 points)

QUESTION 4.

(a).Suppose X has a Poisson distribution with parameter λ=11.44. Find the following.

(i). The mean of X.   (2 points).

(ii).The standard deviation of X.  (2 points).

(b).The number of lightning strikes in a year at the top of particular mountain has a Poisson distribution with parameterλ=2.9.Find the following.

(i). The probability that in a randomly selected year the number of striking is less than or equal to 5.(5 points)

(ii). The probability that in a randomly selected year the number of striking is greater than 5. (3 points)

(c).The probability that a call received by a certain switchboard will be a wrong number is 1 per 100 calls. Use Poisson approximation to the binomial distribution to find the probability that among 120 calls received by the switchboard,

(i).There are no wrong numbers. (2 points)

(ii).There is exactly one wrong number. (2 points)

(iii). There are exactly two wrong numbers. (2 points)

(iv). There are exactly three wrong numbers. (3 points)

(v). There are at most three wrong numbers. (4 points)

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