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- Let X be a discrete random variable taking values {x1, x2, . . . , xn} with probability {p1, p2, . . . , pn}. The entropyof the random variable is defined asH(X) = −Σpilog (pi) Find the probability mass function for the above discrete random variable that maximizes the entropy.2 Law of Large Numbers Define x a random variable with E[x] = µ‚V[x] = o² 0. You can write the unconditional expectation as E[a] = E[a|a 0 =] ≤ k- for k > 0 6. What is the limit of P[|7o – Hol ≥ as No → ∞? You have just proved the law of large numbers (LLN). The LLN states the sample average will grow ever closer to the expected value. In practice, with large samples, you can interchange the expected value and sample mean.Uniform distribution of traffic accidents on a 123 km road (f(x)=C)As it is accepted;a) Determine the value of the constant C,b) Find the additive distribution function (e.d.f =F(x)=?) with respect to the random variable x,c) Find the probability of an accident between 70 and 93 km of the road. (Work 5 digits after the comma.)
- A continuous random variable X has the distribution function if X 3 Find K and f (x), P.d.f.(3) Let X be a random variable with p.d.f. 2k 3x f(x)= ILet X be a random variable and c be a constant. Then Var(X+c) = Var(X), always. True FalseContinuous random variables 6. A charity group raises funds by collecting waste paper. A skip-full will contain an amount, X of other materials such as plastic bags and rubber bands. X may be regarded as a random varibales with probability density funnction. All nmerical values are in units of 100kg. Find the Expected value and Standard deviation.Express your answer in 3 decimal place value f(x)= - c(x-1)(4-x) 0 for 12. The discrete random variable X has the probability function kx, P(X = x) = }k(x – 2), 0, x = 2,4,6 x = 8 otherwise Where k is a constant. (a) Show that k %3D 18 (b) Find the exact value of F(5).Let X and Y be independent exponential random variables with parameters 2 and 3. (a) What is the probability that X + Y > 1. (b) Conditional on X + Y = 1, what is the distribution of X.Let X be a random variable with cumulative distribution function 3 x > 0 x < 0 1 -e F(x) 4 Determine its moment-generating function Use the expectation theorem of the transformation of a random variable3 Let X be a random variable with probability Law P (X = r) = q, K for r = 1,2,3.. . 00, Find the moment generating function and mean of ihe random variable X. Let K+ q = 1.Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON