1. Suppose that X and Y have a discrete joint distribution function as follows: f(x, y) = = xy 36 0 for x = 1, 2, 3; y = 1,2,3 otherwise Determine the probability P(|XY| ≤ 1).
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- The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.Suppose that X is a discrete random variable with P(X = 0) = 0.3, P(X = 1) = 0.2, P(X =2) = 0.15, and P(X = 3) = 0.35. Graph the frequency function and the cumulative distribution function of X.If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
- If X has the Poisson distribution with P(X=1) = 2P(X=2), then P(X ≥ 2) is approximately:If a random variable X has a discrete uniform distribution. fx(x)=1/k for x=1,2,..,k;0 otherwise. Derive P.G.F of X and compute E(2x+1)If the random variable X and Y are independent of each other. X~N(0,1) and the probabilitymass function of Y is P(Y=0)P(Y=1)=1/2. If Fz(z) denotes the cumulative distribution functionof random variable Z=XY, then the number of discontinuites of function Fz(z) is
- The random variable x is known to be uniformly distributed between 1.0 and 1.5. B. Compute P(x=1.25) C. Compute P(1<=x<=1.25) D. Compute P(1.2<x<1.5)Find P(X=4) if X has a Poisson distribution such that 3P(X=1)=P(X=2)If F(x, y) is the value of the joint distribution func-tion of two discrete random variables X and Y at (x, y), show that(a) F(−q, −q) = 0;(b) F(q, q) = 1;(c) if a < b and c < d, then F(a, c) F F(b, d).
- Find k if the joint probability distribution of X, Y, andZ is given by f(x, y, z) = kxyzfor x = 1, 2; y = 1, 2, 3; z = 1, 2.Give any example of a probability distributionover x and y for a classification problem on the real line, so that the Bayesoptimal classifier isf(x) =1, if x ∈[1, 3] ∪[6, 7]−1, otherwiseLet us say that we have jointly distributed random variables X and Y with E(X) = 3,E(Y) = 5, Var (X) = 1, Var(Y) = 3, and Cov (X, Y) = 2.Determine:a) E(2X + Y) b) Var(6X)