an am signal has the equation v(t)=(15+4sin4410^t)sin46.510^6tv find the carrier frequency

an am signal has the equation v(t)=(15+4sin4410^t)sin46.510^6tv find the carrier frequency

Cristiana uses $25,000 to purchase a 25 month CD (certificate of deposit) that pays 2.47% compounded monthly. a. What is her investment worth at the end of 25 months? b. How much interest did she earn? Total: 10 points a. Total: 7 points Breakdown Correct work: 3 points Correct answer: 5 points Sentences explaining reasoning: 2 points b. Total: 3 points Breakdown Correct work: 1 point Correct answer: 1 point Sentences explaining reasoning: 1 point

Cristiana uses $25,000 to purchase a 25 month CD (certificate of deposit) that pays 2.47% compounded monthly. a.

What is her investment worth at the end of 25 months? b. How much interest did she earn?

Total: 10 points a. Total: 7 points Breakdown Correct work: 3 points Correct answer: 5 points Sentences explaining reasoning: 2 points b. Total: 3 points Breakdown Correct work: 1 point Correct answer: 1 point Sentences explaining reasoning: 1 point

Question Find the Fourier transform for the following time functions using the defining integral relationship for the transforms: f(t) = (e^(2t))*[u(t) – u(t-0.5)] g(t) = 6*(e^(-t))*u(t-2)

Question

Find the Fourier transform for the following time functions using the defining integral relationship for the

transforms:

f(t) = (e^(2t))*[u(t) – u(t-0.5)]  

g(t) = 6*(e^(-t))*u(t-2)

An excitable cell is impaled by a micropipette, and a second extracellular electrode is placed close by at the outer-membrane surface. Brief pulses of current are then passed between these electrodes, which may cause it to conduct an action potential. Explain how the polarity of the stimulating pair influences the membrane potential, and subsequently the activity, of the excitable cell.

An excitable cell is impaled by a micropipette, and a second extracellular electrode is placed close by at the

outer-membrane surface. Brief pulses of current are then passed between these electrodes, which may cause it to conduct an action potential. Explain how the polarity of the stimulating pair influences the membrane potential, and subsequently the activity, of the excitable cell.

Q. 1 Suppose X and Y are independent discrete random variables having the Poisson distribution with parameters λ1 and λ2, respectively. Let Z = X +Y, Calculate E[Y | Z = z]. Q. 2 The random variables X and Y have joint probability density function fX,Y (x, y) = (cy x if 0 ≤ y < x ≤ 1, 0 otherwise

EE 351K Probability, Statistics and Stochastic Processes – Spring 2016 Homework 5 Topics: Discrete and Continuous RVs Homework and Exam Grading “Philosophy” Answering a question is not just about getting the right answer, but also about communicating how you got there. This means that you should carefully define your model and notation, and provide a step-by-step explanation on how you got to your answer. Communicating clearly also means your homework should be neat. Take pride in your work it will be appreciated, and you will get practice thinking clearly and communicating your approach and/or ideas. To encourage you to be neat, if you homework or exam solutions are sloppy you may NOT get full credit even if your answer is correct. Q. 1 Suppose X and Y are independent discrete random variables having the Poisson distribution with parameters λ1 and λ2, respectively. Let Z = X +Y, Calculate E[Y | Z = z]. Q. 2 The random variables X and Y have joint probability density function fX,Y (x, y) = (cy x if 0 ≤ y < x ≤ 1, 0 otherwise. (a) Find the constant c, and the marginal densities fX (x) and fY (y). (b) Find P(X +Y ≤ 1). Q. 3 Suppose that the weight of a person selected at random from some population is normally distributed with parameters µ and σ 2 . Suppose also that P(X ≤ 160) = 1 2 and P(X ≤ 140) = 1 4 . Find µ and σ. Also, find P(X ≥ 200). (Use the standard normal table for this problem.) Q. 4 If X has the normal distribution with mean µ and variance σ 2 , find E X 3 (as a function of µ and σ 2 ), without computing any integrals. Q. 5 Let X and Y denote two points that are chosen randomly and independently from the interval [0,1]. Find the probability density function of Z = |X −Y|. Use this to calculate the mean distance between X and Y. Hint: It might be simpler to first calculate P(Z > z). Q. 6 Random variables X and Y are distributed according to the joint probability density function fX,Y (x, y) = ( C if x ≥ 0 and y ≥ 0 and x+y ≤ 1 0 otherwise. Let A be the event {Y ≤ 0.5} and B be the event {Y > X}. (a) Evaluate the constant C. (b) Calculate P(B | A). (c) Find E[X | Y = 0.5] and the conditional probability density function fX|B(x | B). (d) Calculate E[XY]. Q. 7 Let X and Y be independent random variables each uniformly distributed on [0,100]. Find the value of P(Y ≥ X | Y ≥ 12). Q. 8 A coin has an a priori probability X of coming up heads, where X is a random variable with probability density function fX (x) = ( xex , for x ∈ [0,1], 0, otherwise. (a) Find P(Head). Hint: note that according to the hypothesis P(Head | X = x) = x. (b) If A denotes the event that the last flip came up heads, find the conditional probability density function of X given A, i.e., fX|A(x | A). (c) Given A, find the conditional probability of heads at the next flip.

In class, you have learned that the Fourier transform of a rectangular pulse is a sinc function. You have also learned that about 90% of a sinc’s energy is in its main lobe. In this computer exercise, you will explore how the rect function is passed through an ideal low pass filter.

Passing a rectangular pulse through a low pass filter! (75/75 points)

In class, you have learned that the Fourier transform of a rectangular pulse is a sinc function. You have also learned that about 90% of a sinc’s energy is in its main lobe. In this computer exercise, you will explore how the rect function is passed through an ideal low pass filter.

Let x(t) = rect(t) (shown in the above figure) be passed through an ideal low pass filter  where . The following code implements convolution of x(t) and h(t) to get output z(t). We will choose three different values of parameter k and see how k affects the output.

The code below is posted in a MATLAB file named ‘CE3.m’ in files, so you do not have to retype it.

Questions:

  1. Given the definition of , what is the bandwidth (in rads/sec) of the filters ? (5 points each)

Bandwidth is a measure that only includes the positive frequencies

  • What is the bandwidth (in rads/sec) of the main lobe of X(jw) (i.e. the Fourier transform of x(t) )? (10 points)
  • Give hand sketches of the expected frequency response Z(jw) of z(t) for all three values of k (i.e. 0.5, 2, and 10). Hint: Z(jw) = X(jw)H(jw). Next, run the code above and observe how each curve compares to x(t). Using observations from your hand sketches and the code above, why does each z(t) seem distorted? Which is the least distorted (the most like x(t))?

(5 points for each hand sketch, 5 points for each question)

Note: A photograph that includes all three of your sketches is sufficient

  • Write code that can be appended to the above code which will find the energy of z(t) for all three values of k (i.e. 0.5, 2 and 10). Report the percentage of energy of each z(t) as compared to x(t) (i.e. Energy(z(t))/Energy(x(t)). (10 points for code, 5 points for each fraction)

Suppose that a cellular phone costs $4 per month with 10 minutes of use included, and suppose that each additional

Suppose that a cellular phone costs $4 per month with 10 minutes of use included, and suppose that each additional

minute
of use costs $0.50. Suppose the number of minutes you use the phone in a month is well model by a random variable M, such that M Binomial(15, 1/2), this means that the RV M has a binomial pmf with parameters 15 and 1/2 . Find the PMF for the random variable X, corresponding to your bill on a given month.

Topics: functions of random variables, expectation and variance, joint pmfs, conditioning and independence

Question attached. 

A bag contains 100 balls, with an equal number of red, blue, green and yellow balls.

A bag contains 100 balls, with an equal number of red, blue, green and yellow balls.Ball

selection/recording process: Each time I select a ball from the bag, any ball is equally likely to be chosen. I record its

color and then immediately put it back into the bag.

Experiment: I select 10 balls sequentially as above, and record the color configuration, i.e., the number of red, blue, green

and yellow (specific order in which these colors appeared does not matter), for example: (3R;4B;0G;3Y).

Iteration: I perform the experiment above repeatedly and note the configurations. For example, a sequence of configurations

could be: (3;4;0;3); (7;0;0;3); (1;1;1;7); ::: .

See picture for questions.

A spider sits on a wall and repeatedly attempts to catch a fly. Each time it attempts, it succeeds with probability 0:3 (i.e. it misses the fly with probability 0:7). Assume that successive attempts are independent. Let Y be the random variable corresponding to the number of attempts the spider makes to first successfully catch a fly. Describe the sample space, the PMF of the random variable Y; and determine the probability that the spider successfully catches the fly within 5 attempts.

A spider sits on a wall and repeatedly attempts to catch a fly. Each time it attempts, it succeeds with

probability 0:3 (i.e. it

misses the fly with probability 0:7). Assume that successive attempts are independent. Let Y be the random variable corresponding to

the number of attempts the spider makes to first successfully catch a fly. Describe the sample space, the PMF of the random variable

Y; and determine the probability that the spider successfully catches the fly within 5 attempts.