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Chapter 10 Solutions
A First Course in Probability
- Calculate the variance of a random variable X whose characteristic function is (1+ e3it)² f(t) = 4arrow_forwardThe profit for a new product is given by Z = 3X-Y -5. X and Y are independent random variables with Var(X) = 1 and Var(Y) =2 what is the variance of Z?arrow_forwardX is a continuous random variable having p.d.f. f(x) such that f(x) = kr Osx55 otherwise Find : (a) the constant k; (b) P(15XS 2); (c) (d (e) (f) E(X); E(3X + 1); E(X?); Var (X); (g) Var (4X - 5);arrow_forward
- Show that variance o? = (x²) – ((x))²arrow_forward3(8x – x²) Determine the mean and variance of the random variable for f(x) = 0arrow_forwardX is a continuous random variable having p.d.f. f(x) such that f(x) = kx2 Osxs5 otherwise Find : (a) the constant k; (b) P(13XS2); (c) E(X); (d) E(3X + 1); (e) E(X); (f) Var (4X - 5); Var (X); (g)arrow_forwardSuppose that X and Y are independent random variables with finite variances such that E(X) = E(Y). Show that: E[(X-Y)2] =Var(X) +Var(Y)arrow_forward7. Calculate the variance of g (X) = 2X + 3, where X is a random variable with probability distribution X 0 1 2 3 f(x)=1/ 4 1180 31100 1 2 8 First, we find the mean of the random variable 2X + 3.arrow_forwardBy hand solution needed onlyarrow_forwardLet X1, X2, ..., X, be independent random variables and Y = min{X1, X2, ..., Xm}. Fy (y) = 1 – || (1 – Fx,(y)) i=1 (a) A certain electronic device uses 5 batteries, with each battery to have a life that is exponentially distributed with mean of 48 hours and is independent of the life of other batteries. If the device fails as soon as at least one of its batteries fail, what is the expected life of the device?arrow_forwardAn ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.arrow_forwardProb. 3 Let X be a random variable with cumulative distribution function (cdf) given by (1-e-x², x ≥ 0 ={1,- x<0 Find the probability that the random variable X falls within one standard deviation of its mean. Fx (x) =arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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