Statistics and Computational Method Questionnaire

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump withtwo hoses. Let x denote the number of hoses being used on the self-service island at a particular time, and let y denote the
number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying
tabulation.
y
p(x,y)
0
1
2
0
0.10
0.03
0.02
1
0.06
0.20
0.08
2
0.05
0.14
0.32
(a) What is P(X = 1 and Y = 1)?
P(X = 1 and Y = 1) = 2
(b) Compute P(X S 1 and Y S 1).
PIX S1 and Y < 1) = 39 (c) Give a word description of the event {X = 0 and Y + 0}. One hose is in use on one island. At most one hose is in use at both islands. One hose is in use on both islands. At least one hose is in use at both islands. Compute the probability of this event. P(X + 0 and Y = 0) = .74 (d) Compute the marginal pmf of X. 0 1 2 Px(x) .15 34 .51 Compute the marginal pmf of Y. y 0 1 2 Pylv) 21 37 42 Using Py(x), what is P(X < 1)? P(X S 1) = .49 A certain market has both an express checkout line and a superexpress checkout line. Let X, denote the number of customers in line at the express checkout at a particular time of day, and let X, denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X, and X2 is as given in the accompanying table. X₂ 0 1 2 3 0 0.09 0.07 0.04 0.00 1 0.05 0.15 0.05 0.04 X1 2 0.05 0.04 0.10 0.06 3 0.00 0.04 0.04 0.07 0.05 4 0.00 0.01 0.05 = (a) What is P(X2 = 1, X2 = 1), that is, the probability that there is exactly one customer in each line? P(X1 = 1, X2 = 1) = .15 = = (b) What is P(X2 = x2), that is, the probability that the numbers of customers in the two lines are identical? P(X1 = x2) = .41 = (c) Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X1 and X2: O A = {X1 52 + X2 UX2 5 2 + X1} O A = {x1 2 2 + X20X2 2 + X1} O A = {X1 2 2 + X2 UX2 2 2 + x;} O A = {X1 5 2 + X2 UX2 2 2 + X1} 22+ + + + Calculate the probability of this event. P(A) = 23 = (d) What is the probability that the total number of customers in the two lines is exactly four? At least four? P(exactly four) .18 Plat least four) = .46 Lecture 1 - Introduction and Overview.pdf Lecture 2 - Events and Counting Outcomes.pdf Lecture 3 - Probabilistic Models.pdf Lecture 4 - Conditional Probability and Independence.pdf Lecture 5 - Bayesian Theorem and Examples.pdf Lecture 6 - Discrete Random Variables.pdf Lecture 7 - Statistical Mean and Variance.pdf Lecture 8 - Some Distributions of Discrete Random Variables.pdf Lecture 9 - Continuous Random Variables.pdf
Lecture 10 – Statistical Mean and Variance of Continuous RVs
Lecture 11 – Some Distributions of Continuous Random Variables.pdf
Lecture 12 – Transformation of a Random Variable.pdf
Lecture 13 – Joint Distribution of Two Discrete RVs.pdf
Lecture 14 – Joint Distribution of Two Continuous RVS
Lecture 15 – Conditional Probability and Independence
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Question 1
In the following circuit, we want to find i(t) for t>0:
182
w w
www
t=07
1 Ω
gov
+
1 F
2
$ 1H
NI
(a) The steady-state value of i(t) is:
(b) The type of response i(t)
(c) The coefficients of the characteristics equations are s? + Xs +Y = 0, where X:
and Y:
(d) Final solution for i(t) is
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Question 2
9 p
For the following circuit, the coupling coefficient k=1.
K1
0.5
pr
K
th
ad
100 Cosat
to
14
(但
The mutual inductance is
The phasor current passing through coil 1 is
and coil 2 is
The real power delivered by the voltage source is
and reactive power delivered
IS
. Here the power factor is
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The total points are 27, where 2 bonus points are distributed.
9 pts
Question 3
a) Find the transfer function in standard form using
V.(w)
I(W)
b) Zero(s) are @
and pole(s) are at
c) Sketch the Bode plot for magnitude and phase. (The value of H at w=1 is
d) What type of filter can be presented by this transfer function
icLt)
I
R
10
31
Vol
LG
lalu
Assign
XC
х
Submit Answer
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Let X denote the voltage at the output of a microphone, and suppose that X has a uniform distribution on the interval from –1 to 1. The voltage is processed by a “hard limiter” with cutoff values –0.5 and 0.5, so the limiter output is a random variable Y related to X by
Y = X if |X] S 0.5, Y = 0.5 if X > 0.5, and Y = -0.5 if x < -0.5. (a) What is P(Y = 0.5)? (b) Obtain the cumulative distribution function of y. y