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Chapter 10 Solutions
A First Course in Probability
- Assume that the probability that an airplane engine will fail during a torture test is 12and that the aircraft in question has 4 engines. Construct a sample space for the torture test. Use S for survive and F for fail.arrow_forwardIf a binomial experiment has probability p success, then the probability of failure is ____________________. The probability of getting exactly r successes in n trials of this experiment is C(_________, _________)p (1p)arrow_forwardConsider the bivariate distribution for the random variables x and y. The expected value of a linear combination of x and y is E(ax + by) = aE(y) + bE(x) E(ax + by) = aE(x) − bE(y) E(ax + by) = aE(y) − bE(x) E(ax + by) = aE(x) + bE(y)arrow_forward
- Let X be a Poisson random variable such that 2 P(Xe 4P(X 1) +5P(X = 2) %3D 2020/03/om (a) Find its mean and variance. (b) What is the probability of X > 3arrow_forwardLet X be a random variable and a real number. Show that E(X - a)² = varX + (µ − a)² Hereμ = EX is the expected value of the random variable X and varX = E(X - μ)^2 is the variance of the random variable X. Guidance: start from the representation - (X-a)^2 = (X µ + μ- a)^2 and group the right side of the representation appropriately into the form (Z + b)^2, where Z is some random variable and b is a real number and open the square. The task should be solved with the help of the expected value calculation rules.arrow_forwardIf the random variable X follows N (0, 2) and Y = 3X, find the mean and vari- ance of Y.arrow_forward
- ???arrow_forwardLet X be a discrete random variable.(i) Prove thatV ar(X) = E(X2) ? (EX)2.Hint: Use the definition of the variance, the binomial formula, and the facts you know about the expected value.Notice that EX is a real number and can be treated like a constant.(ii) Prove thatV ar(aX + b) = a2V ar(X), for any a, b 2 R.arrow_forwardLet x be random variable with E(x) = 2 and var(x) = 3. Verify that random variable x and the random variable y=-4x+8 are orthogonal. CS Scanned with CamScannerarrow_forward
- Determine ?(?>2).arrow_forwardEXER 6.3 Find the covariance and the correlation coefficient between X and Y, if X and Y are jointly discrete random variables, with joint PMF given by: SHOW SOLUTIONS X\Y 0 1 6 0 28 6 1 28 2 0 333333 28 28 28 2120 28 0arrow_forwardIf the joint probability distribution of X1 and X2 isgiven by f(x1, x2) = x1x236for x1 = 1, 2, 3 and x2 = 1, 2, 3, find(a) the probability distribution of X1X2;(b) the probability distribution of X1/X2.arrow_forward
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