1) Suppose

X ~ N130, 25

. Find; a)

P(X 140) b)

P(X 120)

c)

P(130 X 135)

2) The random variable X is normally distributed with mean 500 and standard deviation 100.

Find; (i)

P(X 400)

, (ii)

P(X 620)

(iii) the 90th percentile (iv) the lower and upper

quartiles. Use graphs with labels to illustrate your answers.

3) A radar unit is used to measure speeds of cars on a motorway. The speeds are normally

distributed with a mean of 90 km/hr and a standard deviation of 10 km/hr. What is the

probability that a car picked at random is travelling at more than 100 km/hr?

4) For a certain type of computers, the length of time bewteen charges of the battery is

normally distributed with a mean of 50 hours and a standard deviation of 15 hours. John

owns one of these computers and wants to know the probability that the length of time

will be between 50 and 70 hours

5) Entry to a certain University is determined by a national test. The scores on this test are

normally distributed with a mean of 500 and a standard deviation of 100. Tom wants to

be admitted to this university and he knows that he must score better than at least 70% of

the students who took the test. Tom takes the test and scores 585. Will he be admitted to

this university?

6) A large group of students took a test in Physics and the final grades have a mean of 70

and a standard deviation of 10. If we can approximate the distribution of these grades by a

normal distribution, what percent of the student; (a) scored higher than 80? (b) should

pass the test (grades60)? (c) should fail the test (grades

7) A machine produces bolts which are N(4 0.09) where measurements are in cm. Bolts are

measured accurately and any bolt smaller than 3.5 cm or larger than 4.4 cm is rejected.

Out of 500 bolts how many would be accepted? Ans 430

8) Suppose IQ ~ N(100,22.5).a woman wants to form an Egghead society which only

admits people with the top 1% IQ score. What should she have to set the cut-off in the

test to allow this to happen? Ans 134.9

9) A manufacturer does not know the mean and standard deviation of ball bearing he is

producing. However a sieving system rejects all the bearings larger than 2.4 cm and those

under 1.8 cm in diameter. Out of 1,000 ball bearings, 8% are rejected as too small and

5.5% as too big. What is the mean and standard deviation of the ball bearings produced?

Ans mean=2.08 sigma=0.2

Over a very long period of time, it has been noted that on Friday's 25% of the customers

at the drive-in window at the bank make deposits. What is the probability that it takes 4

customers at the drive-in window before the first one makes a deposit.

2. It is estimated that 45% of people in Fast-Food restaurants order a diet drink with their

lunch. Find the probability that the fourth person orders a diet drink. Also find the

probability that the first diet drinker of th e day occurs before the 5th person.

3. What is the probability of rolling a sum of seven in fewer than three rolls of a pair of

dice? Hint (The random variable, X, is the number of rolls before a sum of 7.)

4. In New York City at rush hour, the chance that a taxicab passes someone and is

available is 15%. a) How many cabs can you expect to pass you for you to find one that

is free and b) what is the probability that more than 10 cabs pass you before you find

one that is free.

5. An urn contains N white and M black balls. Balls are randomly selected, one at a time,

until a black ball is obtained. If we assume that each selected ball is replaced before the

next one is drawn, what is;

a) the probability that exactly n draws are needed?

b) the probability that at least k draws are needed?

c) the expected value and Variance of the number of balls drawn?

6. In a gambling game a player tosses a coin until a head appears. He then receives $2n ,

where n is the number of tosses.

a) What is the probability that the player receives $8.00 in one play of the game?

b) If the player must pay $5.00 to play, what is the win/loss per game?

7. An oil prospector will drill a succession of holes in a given area to find a productive

well. The probability of success is 0.2.

a) What is the probability that the 3rd hole drilled is the first to yield a productive well?

b) If the prospector can afford to drill at most 10 well, what is the probability that he will

fail to find a productive well?

8. A well-travelled highway has itstraffic lights green for 82% of the time. If a person

travelling the road goes through 8 traffic intersections, complete the chart to find a) the

probability that the first red light occur on the nth traffic light and b) the cumulative

probability that the person will hit the red light on or before the nth traffic light.

9. An oil prospector will drill a succession of holes in a given area to find a productive

well. The probability of success is 0.2.

a) What is the probability that the 3rd hole drilled is the first to yield a productive well?

b) If the prospector can afford to drill at most 10 well, what is the probability that he will

fail to find a productive well?

2. Each day is sunny, cloudy, or rainy. If it is sunny one day, there is a 50% chance that it will be sunny the following day and a 50% chance it will be cloudy. If it is cloudy one day, there is an equal 1 chance of = . 100% that it will be sunny, cloudy, or rainy the following day. If it is rainy one day, there is a 50% chance that it will be cloudy the following day and a 50% chance of rain. (a) Set up a stochastic matrix corresponding to this Markov process. Is the matrix regular? (b) Formulate a system of linear equations for finding the stable distribution for the process. (c) Use the Gauss-Jordan elimination method to solve the system of equations in (b).EEE 304, Communication System I, Spring 2018 (Deadline: 27.03.2018) Generation of random sequences: While deterministic signals such as square pulses, since waveforms, sinuses and cosines are used in specific applications. almost all other real-life signals from econometric series to radar returns, from genetic codes to multimedia signals in consumer electronics are information-bearing random signals. Time sequences: Generate a sequence of 1000 equally spaced samples of a Gauss- Markov process using the recursive relation: 1, = ax,-1 + win = 1,2, .., 1000. Here assume that No = 0. and (w,) is a sequence of independent identically distributed Gaussian random variables. You can use the randn function in MATLAB to generate zero-mean. unit-variance random variables. Below is given a low-pass filter (a>0) excited by a white noise sequence, and the filter has a real pole at 2- a. Plot the output waveform for the following values of a: a= 0.5. a - 0.95. a = 0.995. Comment on the effect of the selection of a on the resulting time sequence. Autocorrelation function: In order to estimate the autocorrelation function of this discrete process, one can apply the formula: Ry(m) = lim - EN- W 2n=1*n In-m = E(X,In-m). However. since we have a finite number of samples an approximation would be Ry(m) = - -mx N-m Zal Andn-mim = 0,1. ....64. Notice we shorten the data and consider only the overlapping samples in the window starting at nel and the window starting at nem+1. The theoretical autocorrelation for such a signal model is also known as: R,(m) = ona", where of is the input process variance. Plot the autocorrelation sequence in three separate graphs for the values of a= 0.5. a = 0.95. a = 0.995, but superpose in each graph the analytical and estimated R,(m) and R, (m) correlation functions. Power Spectrum: Using the Wiener-Khinchine theorem, we can find the power spectrum in two different ways. Then do the following: i. Estimate T(() = (x(/]P. where X (f = ) = EN=x(nje /x*. for a=0.95 and N=512 ii. Estimate ,(f) = [X(f)I again as above, but this time averaging ten estimates. each obtained from a different time sequence. In other words, you must use the recursion In = 0.95x1-1 + Whin = 1,2,....512. ten times, and average the results, i.e., P.(f) = ,|x,(/)|". The estimates are noisy and random, and to improve and smooth the estimates, we must calculate the quantities several times (let us say 10 times) and average them. iii. We also know the analytical value of 4,() = [H(]I 4,(f). where 4w(f) is the power spectral density of the input white noise. hence, 4.(f) = 1, and |1-ae-121 1.9025-1.9cos(271)" Plot the power spectra in these three cases and comment on each of them.Regression Equation in Uncoded Units RESPONSE = 5.839 - 1.345 A/C - 0.096 TR/PA + 0.0868 TI/PR + 0.400 A/C.TR/PA + 0.0189 A/C.TI/PR - 0.0200 TR/PA"TI/PR - 0.0120 A/C*TR/PA*TI/PR Regression Equation in Uncoded Units Response = 9.35 - 2.07 A/C + 0.16 TR/PA + 0.2228 TI/PR . 1.86 A/C'TR/PA . 0.0144 A/C . 0.0519 TR/PA'TV/PR . 0.0544 A/C*TR/PA*TV/PR i want an explination about the numbers in the reggression model our response is kilometer per liter we are studying the affect on the fuel our factor is AC: air conditioning TR/PA: traffic pattren TI/PR: tire pressure 2 cars GMC YOKUN and toyota corolla