If X and Y are two random variables, then the covariance of x+a,Y +b,where a and b are constant is cov(X+a,Y+b) =cov(X,Y)=Cxy
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- 2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.7 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 3 Food C 1 1 1Suppose X and Y are random variables with E[XY ] = 6, E[Y ] = 4 and E[X] = 5 Find Cov(X; Y )Let X1, X2, . . . are independent indicator variables with different probabilities of success. That is, P(Xi = 1) = pi. Define Yn = X1 + X2 + . . . + Xn. Find E(Yn), V ar(Yn) and coefficient of variation of Yn
- Suppose we have two random variables X and Y. For a, b > 0, a2+b2=13. Var(aX) = 16 Var(bY) = 36 Var(X - Y) = 2 Cov(2X, Y) = 6 1) What is ab? 2) What is Corr(X, Y)?1- The number of items produced in a factory during a week is known to be a randomvariable with mean 50● Using Markov's inequality, what can you say about the probability that this week'sproduction exceeds 75?● If the variance of one week's production is equal to 25, then using Chebyshev'sinequality, what can be said about the probability that this week's production isbetween 40 and 60?2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).
- Let X and Y be random variables. Suppose Var(X) = 1.6, Var(Y) = 1.5, and Cov(X, Y) = 0.1. Let Z = -1.2X + 1.2Y + 4.7. Calculate Var(Z).For the random variables X,Y Cov(X,Y) = -0.9 if Z=3-X then what is Cov(Z,Y)=???Let X1, X2, X3 be random variables such that V ar(X1) = 5, V ar(X2) = 4,V ar(X3) = 7, cov(X1, X2) = 3, cov(X1, X3) = −2 and X2 and X3 are independent. Findthe covariance between Y1 = X1 − 2X2 + 3X3 and Y2 = −2X1 + 3X2 + 4X3
- Suppose Xis a random variable with E(X) = 2, and E(X2) = 5. Compute the following quantities: a) Var(X) b) E(X−1)2 c) Var(−X−3) d) E(5X+ 4) PARTS A,B,C ARE ANSWERED ALREADY, JUST D PLEASEShow that if X is not a deterministic random variable, then H(X) is strictly positive. What happens to the probabilities if a random variable is non-deterministic?Suppose Xn is an IID Gaussian process, withµX[n]=1, and σ2 X[n]=1Now, another stochastic process Yn = Xn − Xn−1. Please find:(a) The mean µY (n).(b) The variance σ2Y (n).(c) The auto-correlation RY (n, k)
