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- how do I use markov's and chepyshev's inequalities to obtain bounds for the probability of having an event A or B given the expected value and the variance?I roll a fair die twice and obtain two numbers: X1 = result of the first roll, and X2 = result of the second roll. Write down the sample space S, and assuming that all outcomes are equally likely (because the die is fair), find the probability of the event A defined as the event that X1 + X2 = 8.Let X and Y be any two random variables and let a and b be any two real numbers. Then var(ax + bY) = a² var(X) + b?var(Y) + 2ab cov(X,Y). False True
- let X(n,p) denotes binomial random variable with parameters n and p. Show that X(n+1,p) >st ( greater than equal ) X(n,p)Q1:Show that: If X1, X2, , Xn are independent random variables and X = X1 + X2 ++Xn, then Q2: What is the expectation and the variance of RV X, where X represents the out come throwing a die? Q3: Find the expectation and the variance of X, where X is binomial random variable X - Binomial (n, p), Var(X)? 1 Chapter 3 Let X be a discrete random variable with range Rx = {1, 2, 3, ...}. Q4: Suppose the PMF of X is given by 1 for k = 1, 2, 3, ... 2k Px (k) = a) Find and plot the CDF of X, Fx (a). b) Find P(1 < X < 3). Chapter 3 Functions of Random Variables Q5: Example. Let Rx = {0, T T 37 , such that 4'2 4. Find E[sin(X)]. Q6: A machine produces a defected items with a probability of 0.1. What is the probability that in a sample of three item will have at most one item defected? Q7: For any independent X, and Y random variables, E[XY]=E[X]E[Y]. Show that? Q8: Suppose RV X has the following PMF: p(0)=0.2, p(1)=0.5, p(2)=0.3. Find E[X], E[X^2], E[X]^2. 3 Chapter 3In a digital communication channel, assume that the number of bits received in error can be modeled by a binomial random variable, and assume that the probability that a bit is received in error is 0.0001. If 150,000 bits are transmitted, what is the probability that errors are between 10 and 18? How to compute P(X=10)= (150000c10)(0.0001)10(1-0.0001)150000-10
- (3) Let X be a random variable with p.d.f. 2k 3x f(x)= ISuppose that X is a random variable * ? having the following m.g.f., Find Var(x) 7t_e4t M, (t) = eT 173 O 5 O 413 O 3148 O 55 OFor the probability distribution of a discrete random variables, the sum of the probabilities of all possible values of x is always