Practice Chapter 61. Let X1 ~ N(10,2) and X2 ~ N(-5,5) with X1 and X2 independent. Also let X = 2X1

Practice Chapter 61. Let X1 ∼ N(10,2) and X2 ∼ N(−5,5) with X1 and X2 independent. Also let X = 2X1 +3X2.(a) What is the distribution of X? (State its name and parameters.)(b) Find P(X > −2).2. The elongation of a steel bar under a particular load has been established to be normallydistributed with a mean of µ = 0.05 and a standard deviation of σ = 0.01. Find the probabilitythat the elongation is(a) above 0.1 inch;(b) between 0.025 and 0.065 inch.3. If a technician does not encounter any hardware problems, the time he requires to assemble acomputer follows a normal distribution with a mean of 30 minutes and a standard deviation of 3minutes. Let T be the time in which he assembles a computer.(a) Find the probability that it will take him more than 36 minutes to assemble a computer giventhat he does not encounter hardware problems.(b) When he encounters hardware problems the time to assemble a computer has a mean of 50minutes and a standard deviation of 7 minutes. Find the probability that it will take him morethan 36 minutes to assemble a computer given that he encounters hardware problems.(c) Suppose that he encounters a hardware problem 10% of the time. If it took him more than 36minutes to assemble a computer, what is the probability that he encountered a hardwareproblem?4. Suppose that the time between calls from your aunt Debbie has an exponential distributionwith a mean time of 3 days.(a) If you just received a call from her, what is the probability that you will receive the next callwithin the next 2 days?(b) You realize that you have not received a call in at least 2 days. What is the probability thatshe will not call you within the next two days?(c) Find the probability that you will receive two calls in less than 3 days. Assume independencebetween the calls.5. Let X ∼ Gamma(2, 1) and Y ∼ Gamma(3, 1) with X and Y independent.(a) Find E((X − Y ) 2 )(b) Find Var((X + 1) Y ).6. Let X1 ∼ Poisson(10) and X2 ∼ Poisson(5), with X1 and X2 independent and set X =X1 + X2. (a) What is the distribution of X?(b) Find P(X > 18).

 

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