H62NUM Probability
A good way of remembering the AND, OR notation
AND = n = looks like an 'n' - sound it out "aNNNd" OR = u = looks like a 'u' - imagine someone verbally 'doing a u-turn' like: "ORRRR we could do this...."
P(system available) = P(tx working)xP(chworking)xP(rx working) therefore 1-P(systemunavailable) = [1-Pdtx)]x[1-Pdch]x[1-Pdrx] so: P(systemunavailable) = 1-([1-Pdtx)]x[1-Pdch]x[1-Pdrx]) These brackets all multiply out to: P(systemunavilable) = Pdtx + Pdch + Pdrx - (PdtxPdrx + PdtxPdch + PdchPdrx) + PdtxPdchPdrx The system should be moderately designed, so we can approximate the products terms to zero, leaving just: Pdsystem = Pdtx + Pdch + Pdrx MDTx = 365x24x60xPdx, 360x24x60 'cancels'? so: MDTsystem = MDTtx + MDTch + MDTrx
Assuming independence of each element, what is the equation for P(system unavailable) in a comms link? What is the expanded version? What approximations can we make to simply this and why?
P(E1 | E2)=P(E1nE2)/P(E2) I.e., the probability of Event E1 based on Event E2 occuring is given by the probability of both E1 AND E2 divided by the probability of only E2 occuring.
Conditional Probability Formula for two events, E1 and E2
PY(y) = pX(x).|dx/dy| >We know the pdf, pX(x) >Get the transformation of X into Y, eg Y=X^3 > Work out range of new pdf based on this. >Get dy/dx of transformation of X into Y, so in this case dy/dx= 3X^2 >Sub pX(x) and dy/dx (reciprocal of dy/dx we just worked out) into the transformation equation. >sub in x in terms of y to the result. >This is the new pdf pY(y) for the new range we worked out a couple of steps ago. >manipulate P(whatever) so its workable (ie from P(X^2>=2), square root both sides of inequality to get P(X>=root2)
Give the equation and explain the procedure for transforming a known continuous pdf pX(x) to a new pdf pY(y).
Fx(x) = Fz[(x-u)/sigma] i.e., plug x into: (x-u)/sigma to get the value needed for lookup in the erf table. If negative, we have F(-z)=1-F(z), so find result using 1-erf(|z|)
Give the expression used for transformation of normally distributed random variable X~N(u,sigma^2) to standardised normally distributed random variable Z~N(0, 1).
Peb = P(1|0).P(0) + P(0|1).P(1) The probability of a 1 being detected, given a 0 transmitted, multiplied by the probability of a 0 being transmitted. PLUS the probability of a 0 being detected, given a 1 transmitted, multiplied by the probability of a 1 being transmitted. Usually 1 and 0 are equiprobable so P(1)=P(0)=0.5
Give the general expression for binary error probability (BER).
P(X=r) = (ncr).p^r.q^(n-r) q = 1-p ncr = "n choose r", r= number of successes/results of interest (integer), n=total number of trials/outcomes. ncr is 'The binomial coefficient', can be obtained by the nth row and rth column of Pascal's triangle (excluding the 1 at the top of the triangle). ncr = n!/(n-r)!r! also. p=probability of what we define as a 'success', a constant The Binomial distribution is used in situations where we have two possible outcomes of a trial, success/fail, pass/reject, etc. Mean = u = np Variance = sigma^2 = npq
Give the general form of the Binomial distribution and define each term. When sort of problems is this distribution used for? What is the mean and variance?
https://www.dropbox.com/s/uhier2z8js1n6nb/Photo%2028-05-2017%2C%2010%2048%2033.jpg?dl=0
Let D=the decision threshold at the reciever of a comms system. What is the binary error probability?
P(E1)+P(E2)+P(E3) = P(E1uE2uE3)-P(E1nE2nE3 )+ (P(E1nE2)+P(E1nE3)+P(E2nE3))
Probability Addition Rule for three events
P(E1)+P(E2) =P(E1nE2)+P(E1uE2) I.e, the probability of event 1 plus the probability of event 2, equals the probability of E1 AND E2 both occurring plus the probability of either E1 OR E2 occurring.
Probability Addition Rule for two events
http://www.sosmath.com/calculus/tayser/tayser01/img5.gif Each derivative term (v(T), dv(t)/dt, d''v(t)/dt''... etc) corresponds to coefficients a,b,c... in the interpolating polynomial. (the local approximation function)
Recall the general form of the taylors series (you will need to remember this in the exam) How does each term link with the numerical schemes?
planck constant = 6.63 x10^-34 hf = planck constant x optical frequency f = energy (Joules) bit period T = 1/data_rate Average bit energy = Paverage x bit_duration_T Energy = power x time Mean number of photons per bit = average_bit_energy/hf
Show how to calculate photon energy hf, and the mean number of photons in one bit in an optical comms system
https://www.dropbox.com/s/17331spahpscppi/Photo%2028-05-2017%2C%2010%2050%2038.jpg?dl=0
Show how we can calculate the optimal threshold point, Dopt, for a receiver, to get minimum BER. How is this used to set specific Peb values?
https://www.dropbox.com/s/dc4aqz6vqrjmhn4/Photo%2028-05-2017%2C%2015%2033%2013.jpg?dl=0
State the cumulative distribution function for a DISCRETE random variable.
p(x) = lamda.e^(-lamda.x) for x>=0 mean = u = 1/lamda variance = sigma^2 = 1/lamda^2 cdf = 1-e^(-lamda.x) cdf^1 (to find x from y) = [-ln(1-X)]/lamda
State the exponential distribution, its mean, variance, cdf, and inverse function (ie where y is the argument) for the cdf.
https://www.dropbox.com/s/wbp9ogsce0za45m/Photo%2027-05-2017%2C%2013%2054%2053.jpg?dl=0 X=number of events occuring in a time interval T Lamda = the mean rate at which events occur ie we have a mean rate of 1000 data packets a second in a comms system - what is the probability of 1 data packet arriving in a 1ms interval? Then lamda = 1000, T=0.001, mu = lamda.T = 1. We use this distribution when the random variable X takes on integer values, and the events we are counting are INDEPENDENT and occurring at a constant (mean) rate.
State the general formula for the Poisson Distribution, the mean, and the Variance. Define each term in the distribution. When is this distribution used?
https://www.dropbox.com/s/ymfhd4nvbn0gyoo/Photo%2027-05-2017%2C%2013%2047%2016.jpg?dl=0
State the general formula for the Variance, V(X)
https://www.dropbox.com/s/mwy54l9eplb7fb6/Photo%2027-05-2017%2C%2013%2035%2010.jpg?dl=0
State the general formula for the expectation of X, E(X), for both the discrete and continuous case.
https://www.dropbox.com/s/2y3rrut5avl5mze/Photo%2027-05-2017%2C%2013%2036%2010.jpg?dl=0
State the general formula for the expectation of f(x), E(f(x)), for both the discrete and continuous case.
When n is very large but P is very small - we can approximate using a Poisson of mean = u = np. Generally acceptable for n>30, np<10.
Under what conditions can a Binomial distribution be approximated by Poisson? What key parameter in the Poisson is different?
PROBABILITY is simply the proportion (fraction, ratio) of a particular outcome(s) we are interested in versus the amount of total instances/experiments/trials.
What actually is the definition, in real person human words, of 'PROBABILITY'?
A random variable (a possible result, an outcome) that has an equal probability of occurring as all the other outcomes in the sample space. An example would be a dice - 6 faces (6 possible outcomes, so a sample size of 6) and each outcome (face) has an equal probability of 1/6th. pdf= 1/(b-a) cdf = x/(b-a) mean = (a+b)/2 variance = (b-a)^2/12
What is a Uniform Random Variable? Give the distribution, cdf, mean and variance.
A selection of outcomes that have particular properties. An event is like a subset of the sample space. So for tossing a coin, the sample space is (heads, tails) as this is the whole set of possible outcomes. The sample space could be split into two events, one being Heads (E1) and the other tails (E2)
What is an 'event'?
Imagine we write out a 'matrix' of the sample space (all our possible outcomes). We 'circle' or highlight all results in the sample space that satisfy event E1 and Event E2 respectively. We then find: the number of 'overlaps' of event E1 onto event E2 divided by the set size of event E2
What is an intuitive explanation of how to calculate conditional probability for 2 events E1 and E2?
The set of all the possible outcomes (results) of one iteration of an 'experiment' or trial
What is sample space (S)?
Q = (u1-u0)/(sig1+sig0) Q=6 gives BER 10^-9 (lab) Q=7 gives BER 10^-12 (commercial)
What is the Q from the simplified Peb equation? What Q values give typical lab and commercial BER values?
https://www.dropbox.com/s/5bi0znuf71vf186/Photo%2029-05-2017%2C%2016%2051%2028.jpg?dl=0
What is the algorithm for the simpsons rule?
https://www.dropbox.com/s/5bi0znuf71vf186/Photo%2029-05-2017%2C%2016%2051%2028.jpg?dl=0
What is the algorithm for the trapezium rule?
Width (delta) x average height average height = (a+b)/2
What is the area of a trapezium?
https://www.dropbox.com/s/rzsjjyx3edywnuo/Photo%2028-05-2017%2C%2010%2048%2051.jpg?dl=0
What is the general formula for P(X>x) and P(X<x) in a normal distribution?
https://www.dropbox.com/s/en25f94p12ehcln/Photo%2029-05-2017%2C%2020%2046%2048.jpg?dl=0
What is the standard convolution formula for the pdf of Z? (When z is the sum of two random variables px1(x) and px2(x))
https://www.dropbox.com/s/ymgddzc1jpfr3b1/Photo%2029-05-2017%2C%2017%2026%2046.jpg?dl=0
What is the standard form of the Lagrange polynomials? Write the first 2 or 3 terms.
number_of_possible_values to the power of number_of_iterations for example number of possible voltage levels in a digital system is 2^n. A bit can only have binary values 1 or 2, so number_of_possible_values=2. n is the number of bits (the number of possible 1s or 0s), so for a 3 bit system we would have 2^3 = 8 possible outcomes. for a dice, there are 6 faces so number_of_possible_values=6. if we throw 3 dice then the size of the sample space (the number of possible outcomes that may happen) is 6^3=216
What simple calculation gives the size of the sample space? (discrete random variable problems)
V(X) = E[(X-u)^2] = E(X^2)-2uE(X)+u^2 = E(X^2)-[E(X)]^2 E(X^2) = integral -inf to +inf of: x^2.p(x) dx
Write the steps in the derivation of V(X) How do we calculate E(X^2)?
https://www.dropbox.com/s/85j49tbz9vdaufn/18835241_1356084451150890_1468042043_n.jpg?dl=0
Describe the steps in deriving the trapezium rule