COMP 211 DATA AND SYSTEM MODELING (PROB/STAT) Spring 2012 Assignment #2 Due: Monday, 5pm, 4/16/2012 Total points: 200 (each question 20 points) Please submit a softcopy (in PDF format) of your assignment to WebCT before the deadline. Late penalty: within 24 hours after the deadline: ‐20%; after 24 hours: 0 point. Question 1: [20 points] A film-coating process produces films whose thickness are normally distributed with a mean of 110 microns and a standard deviation of 10 microns. For a certain application, the minimum acceptable thickness is 90 microns. (a) What proportion of films will be too thin? (b) To what value should the mean be set so that only 1% of the films will be too thin? (c) If the mean remains at 110, what must the standard deviation be so that only 1% of the films will be too thin? Question 2: [20 points] If a resistor with resistance R ohms carries a current of I amperes, the potential difference across the resistor, in volts, is given by V=IR. Suppose that I is lognormal with parameters μI =1 and σI2 = 0.2, R is lognormal with parameters μR =4 and σR2 = 0.1, and that I and R are independent. (a) Show that V is lognormally distributed , and compute the parametersμV and σV2 (Hint: ln V = ln I + ln R) (b) Find P(V < 200) (c) Find P(150≦V≦300) (d) Find the mean of V (e) Find the median of V (f) Find the standard deviation of V Question 3: [20 points] The number of traffic accidents at a certain intersection is thought to be well modeled by a Poisson process with a mean of 3 accidents per year. (a) Find the mean waiting time between accidents. (b) Find the standard deviation of the waiting times between accidents. (c) Find the probability that more than one year elapses between accidents. (d) Find the probability that less than one month elapses between accidents. (e) If no accidents have occurred within the last six months, what is the probability that an accident will occur within the next year? Question 4: [20 points] If T is a continuous random...

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NormalDistributionNormaldistribution is a statistics, which have been widely applied of all mathematical concepts, among large number of statisticians. Abraham de Moivre, an 18th century statistician and consultant to gamblers, noticed that as the number of events (N) increased, the distribution approached, forming a very smooth curve.
He insisted that a new discovery of a...

...
STAT 200
Section 7983
Sping 2012
`
Quiz #2
Please answer all 6 big questions. The maximum score for each question is posted at the beginning of the question, and the maximum score for the quiz is 60 points. Make sure your answers are as complete as possible and show your work/argument. In particular, when there are calculations involved, you should show how you come up with your answers with necessary tables, if applicable. Answers that come straight from...

...NormalDistribution
It is important because of Central Limit Theorem (CTL), the CTL said that Sum up a lot of i.i.d random variables the shape of the distribution will looks like Normal.
Normal P.D.F
Now we want to find c
This integral has been proved that it cannot have close form solution. However, someone gives an idea that looks stupid but actually very brilliant by multiply two of them.
reminds the...

...NORMALDISTRIBUTION
1. Find the
distribution:
a.
b.
c.
d.
e.
f.
following probabilities, the random variable Z has standard normal
P (0< Z < 1.43)
P (0.11 < Z < 1.98)
P (-0.39 < Z < 1.22)
P (Z < 0.92)
P (Z > -1.78)
P (Z < -2.08)
2. Determine the areas under the standard normal curve between –z and +z:
♦ z = 0.5
♦ z = 2.0
Find the two values of z in standard normaldistribution so that:
P(-z < Z...

...standard deviation = square root of variance = sqrt(846) = 29.086 4. If we have the following data
34, 38, 22, 21, 29, 37, 40, 41, 22, 20, 49, 47, 20, 31, 34, 66 Draw a stem and leaf. Discuss the shape of the distribution. Solution: 2 3 4 5 6 | | | | | 219200 48714 0197 6
This distribution is right skewed (positively skewed) because the “tail” extends to the right. 5. What type of relationship is shown by this scatter plot?
45 40 35 30 25 20 15 10 5 0 0 5...

...NormalDistribution:- A continuous random variable X is a normaldistribution with the parameters mean and variance then the probability function can be written as
f(x) = - < x < , - < μ < , σ > 0.
When σ2 = 1, μ = 0 is called as standard normal.
Normaldistribution problems and solutions – Formulas:
X < μ = 0.5 – Z
X > μ = 0.5 + Z
X = μ = 0.5
where,
μ = mean
σ...

...decimal places)
2. Find the value of z if the area under a Standard Normal curve
a) to the right of z is 0.3632;
b) to the left of z is 0.1131;
c) between 0 and z, with z > 0, is 0.4838;
d) between -z and z, with z > 0, is 0.9500.
Ans : a) z = + 0.35 ( find 0.5- 0.3632 = 0.1368 in the normal table)
b) z = -1.21 ( find 0.5 – 0.1131 = 0.3869 in the normal table)
c ) the area between 0 to...

...random will be between 19 and 31 is about 0.95. This area (probability) is shown fir the X values and for the z values.
σ = 3 0.95 σ = 1 0.95
X
19 25 31 -2 0 +2
Normal curve showing Standard normal curve showing
area between 19 and 31 area between -2 and +2
Entry to a certain University is determined by a national test. The scores on this test are normally distributed with a mean of 500...

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