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1.. If X1 and X2 are independent random variables with E(X1)=2, E(X2)=5 Var(X1)=2, and Var(X2)=10, and Y=3X1+5X2, determine the following: E(Y), Var(Y).2. The time between surface finish problems in galvanizing process is exponentially distributed with a mean of 40 hours. A single plant operates three galvanizing lines that are assumed to operate independently.a. What is the probability that none of the lines experiences a surface finish problem in 40 hours of operation.b. What is the probability that all three lines experience a surface finish problem between 20 and 40 hours of operation.3. In a digital communication channel, assume that the number of bits recevied in error can be modeled by a binomial random variable, and assume that the probabilitz that a bit is recived in error is 10^-5 . If 16 million bits are trasmitted, what is the probability that more than 150 errors occur?1. The life of a particular type of dry-cell battery is normally distributed with mean 600 days and standard deviation 60 days. What fraction of these batteries would be expected to survive beyond 680 days? What fraction would be expected to fail before 550 days? 2. Suppose that the random variable X has the following density function f(x)=x/8 if 0<=x<=4otherwiseGive the distribution function of X.What is the distribution function of Y if Y=2X+8?Give the value of P(0.5<=x<=5) ?

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