230
Seat IE # ________ Please read these directions.
Name _____________________
Closed book and notes. 60 minutes. Covers through the normal distribution, Section 4.7 of Montgomery and Runger, fourth edition. Cover page and four pages of exam. Page 8 of the Concise Notes. No calculator. No need to simplify beyond probability concepts. For example, unsimplied factorials, integrals, sums, and algebra receive full credit. Throughout, f denotes probability mass function or probability density function and F denotes cumulative distribution function. A true-or-false statement is true only if it always true; any counter-example makes it false. For one point, write your name neatly on this cover page and circle your family name.
Score ___________________________
Exam #2, March 8, 2011
Schmeiser
IE 230 Probability & Statistics in Engineering I
Name _______________________
Closed book and notes. 60 minutes. For statements ag, choose true or false, or leave blank. (three points if correct, one point if left blank, zero points if incorrect) (a) T F The continuous-uniform family of distributions is a special case of the discrete-uniform family. (b) T F Scheduled arrivals, such as to physicians ofce, are naturally modeled with a Poisson process. (c) T F Consider a discrete random variable X with probability mass function f X . Then f X (c ) dc = 1.
(d) T
F Consider a random variable X with Poisson distribution with mean 2 1/2 = 3 arrivals. Because X = , the units of X are arrivals . F If Z is a standard-normal random variable, then P(Z = 0) = 0.
(e) T
(f) T F If Z is a standard-normal random variable, then f Z (0) = 0, where f Z is the probability density function of Z . (g) T F If Z is a standard-normal random variable, then FZ (3.2) = FZ (3.2), where FZ is the cdf of Z . 2. Suppose that the random variable X has the discrete uniform distribution over the set {1, 2,..., 10} and that the random variable Y has the continuous uniform distribution over the set [1, 10]. For statements ae, choose true or false. (a) (b) (c) (d) (e) (3 points) T (3 points) T (3 points) T (3 points) T (3 points) T F F F F F
E(X ) = E(Y ). V(X ) = V(Y ). FX (5.5) = FY (5). f X (5.5) = f Y (5). P(X = 5) = f X (5).
Exam #2, March 8, 2011
Page 1 of 4
Schmeiser
IE 230 Probability & Statistics in Engineering I
Name _______________________
3. (from Montgomery and Runger, 450) The time until recharge for a battery in a laptop computer under common conditions is normally distributed with mean 260 minutes and a standard deviation of 50 minutes. (a) (8 points) Sketch (well) the corresponding normal pdf. Label and scale both axes.
(b) (6 points) In your sketch, show the probability that a randomly selected recharge is less than three hours. (c) (5 points) State the numerical value to as much precision as you can determine (given that you dont have access to a normal table).
(d) (4 points) Give at least one reason why the time until recharge cannot possibly have a normal distribution.
Exam #2, March 8, 2011
Page 2 of 4
Schmeiser
IE 230 Probability & Statistics in Engineering I
Name _______________________
4. Consider the binomial pmf f X (x ) = Cxn p x (1 p )n x for x =0, 1,..., n and zero elsewhere. (a) (5 points) Explain, in words, the origins of p x . Include any assumptions that are required for your explanation.
8 (b) (3 points) Evaluate C 3 .
(c) (5 points) Explain, in words, the phrase "and zero elsewhere".
5. Consider Question 1, which is composed of seven true-false question. Suppose that a clueless student is taking this exam and answers all true-false questions by ipping a coin, with heads yielding "true" and tails yielding "false". Let X denote the number of questions answered correctly. (a) (6 points) Choose an appropriate distribution for X . Include the family name, parameter values, and probability mass-or-density function f X .
(b) (3 points) Consider another clueless student who leaves all seven questions blank. Determine this students expected number of points?
(c) (3 points) Again consider the student leaves who all seven questions blank. Determine the standard deviation of this students number of points.
Exam #2, March 8, 2011
Page 3 of 4
Schmeiser
IE 230 Probability & Statistics in Engineering I
Name _______________________
6. (from Montgomery and Runger, 411) Suppose that the cumulative distribution function of the random variable X is FX (x ) = 0.2x for 0 x c . Assume that values outside the interval [0, c ] are not possible. (a) (5 points) Sketch FX over the entire real-number line. Label and scale both axes.
(b) (5 points) Determine the value of c .
(c) (5 points) Write f X . Be complete.
Exam #2, March 8, 2011
Page 4 of 4
Schmeiser
IE 230 Probability & Statistics in Engineering I
Discrete Distributions: Summary Table random variable X distribution name general x 1, x 2, . . . , xn range probability
Name _______________________
expected value
n
variance
mass function P(X = x ) = f (x ) = f X (x )
n
xi f (xi )
i =1
(xi )
i =1 2
2
f (xi )
= = X = E(X )
n
= = X = V(X )
2
2
= E(X )
n
2 2
X
discrete uniform
x 1, x 2, . . . , xn
1/n
xi / n
i =1
[ xi / n ]
2 i =1
X "# successes in 1 Bernoulli trial" "# successes in n Bernoulli trials" "# successes in a sample of size n from a population of size N containing K successes" "# Bernoulli trials until 1st success" "# Bernoulli trials until r th success" "# of counts in time t from a Poisson process with rate "
equal-space uniform indicator variable binomial
x = a ,a +c ,...,b where x = 0, 1 x = 0, 1,..., n
1/n n = (b a +c ) / c x 1x p (1p )
n Cx
a +b 2 p where
c (n 1) 12 p (1p ) p = P("success") np (1p ) p = P("success") (N n ) np (1p ) (N 1) p =K /N
2
2
p (1p )
x
n x
np where
hypergeometric (sampling without replacement) geometric
x= (n (N K )) , ..., min{K , n } and integer x = 1, 2,... x = r , r +1,...
+
Cx Cn x / Cn
K
N K
N
np where
p (1p )
x 1 Cr 1
x 1
1/p
x r
(1p ) / p
2
negative binomial Poisson
p (1p )
x
r
where r /p where
p = P("success") 2 r (1p ) / p p = P("success") = t
x = 0, 1,...
e
/ x!
where
x
Result. For x = 1, 2,..., the geometric cdf is FX (x ) = 1 (1 p ) . Result. The geometric distribution is the only discrete memoryless distribution. That is, P(X > x + c | X > x ) = P(X > c ). Result. The binomial distribution with p = K / N is a good approximation to the hypergeometric distribution when n is small compared to N .
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IE230
CONCISE NOTES
Revised August 25, 2008
Continuous Distributions: Summary Table random distribution range variable name cumulative distrib. func. probability density func. dF (y ) dy y =x = f (x ) = f X (x ) 1 b a expected value
variance
2
X
general
(, ) P(X x ) = F (x ) = FX (x ) x a b a
xf (x )dx (x )
f (x )dx
2
= = X = E(X ) a +b 2
= = X = V(X ) 2 2 = E(X ) (b a ) 12
2 2
2
X
continuous [a , b ] uniform
X
triangular
[a , b ]
2(x d ) a +m +b (b a ) (m a )(b m ) (x a ) f (x ) / 2 if x m , else (b a )(m d ) 3 18 1(b x ) f (x )/2 (d = a if x m , else d = b )
1 x
2
sum of random variables time to Poisson count 1 time to Poisson count r lifetime
normal (or Gaussian)
(, ) Table III
e
2
2
2
exponential [0, )
1e
x
e
x
1/
1/
2
Erlang
[0, )
k =r
e
x
(x )
k
x
r r 1 x
e
k!
(r 1)!
r /
r /
2
gamma
[0, ) [0, )
numerical 1e
(x /)
x x
r r 1 x
e
(r )
1 (x /)
r / (1+
r / 1
2
2
lifetime
Weibull
e
)
(1+
2
)
2
Denition. For any r > 0, the gamma function is (r ) = x
0
r 1 x
e
dx .
Result. (r ) = (r 1)(r 1). In particular, if r is a positive integer, then (r ) = (r 1)!. Result. The exponential distribution is the only continuous memoryless distribution. That is, P(X > x + c | X > x ) = P(X > c ). Denition. A lifetime distribution is continuous with range [0, ). Modeling lifetimes. Some useful lifetime distributions are the exponential, Erlang, gamma, and Weibull.
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