Planetary Communications, Inc., intends to launch a satellite that will enhance
ID: 438782 • Letter: P
Question
Planetary Communications, Inc., intends to launch a satellite that will enhance reception of television programs in Alaska. According to its designers, the satellite will have an expected life of six years. Assume the exponential distribution applies. T / MTBF e-T / MTBF T / MTBF e-T / MTBF T / MTBF e-T / MTBF 0.10 .9048 2.60 .0743 5.10 .0061 0.20 .8187 2.70 .0672 5.20 .0055 0.30 .7408 2.80 .0608 5.30 .0050 0.40 .6703 2.90 .0550 5.40 .0045 0.50 .6065 3.00 .0498 5.50 .0041 0.60 .5488 3.10 .0450 5.60 .0037 0.70 .4966 3.20 .0408 5.70 .0033 0.80 .4493 3.30 .0369 5.80 .0030 0.90 .4066 3.40 .0334 5.90 .0027 1.00 .3679 3.50 .0302 6.00 .0025 1.10 .3329 3.60 .0273 6.10 .0022 1.20 .3012 3.70 .0247 6.20 .0020 1.30 .2725 3.80 .0224 6.30 .0018 1.40 .2466 3.90 .0202 6.40 .0017 1.50 .2231 4.00 .0183 6.50 .0015 1.60 .2019 4.10 .0166 6.60 .0014 1.70 .1827 4.20 .0150 6.70 .0012 1.80 .1653 4.30 .0136 6.80 .0011 1.90 .1496 4.40 .0123 6.90 .0010 2.00 .1353 4.50 .0111 7.00 .0009 2.10 .1255 4.60 .0101 2.20 .1108 4.70 .0091 2.30 .1003 4.80 .0082 2.40 .0907 4.90 .0074 2.50 .0821 5.00 .0067 -------------------------------------------------------------------------------- Determine the probability that it will function for each of the following time periods: a. More than 9 years. T e-T/MTBF >9 -------------------------------------------------------------------------------- b. Less than 12 years. T e-T/MTBF <12 -------------------------------------------------------------------------------- c. More than 9 years but less than 12 years. T e-T/MTBF 921Explanation / Answer
Since the mean of the exponential is 30, the pdf is f(x)=(1/30)e^(-x/30) Integrating (0 to x), we find that the cdf is F(x) = 1-e^(-x/30) Remember the cdf is the probability the unit will fail before x. I'm not going to do your homework for you, but I'll do (1) from each set. A1) The probability that the unit lasts at least 39 months is 1-(the probability the unit will fail before 39 months) 1-F(39) e^(-39/30) [I don't have a calculator, so I'll leave that to you.] B1) This is the exact definition of the cdf, so just calculate F(33): F(33) 1-e^(-33/30) C1) We've been calculating the probability/percentage this whole time. It's what the cdf equals. So, to find the length before 50 percent fail, we want to find x such that F(x)=0.5 F(x) = 0.5 1-e^(-x/30) = 0.5 e^(-x/30) = 0.5 [subtract 1 from both sides and multiply by -1] -x/30 = ln(0.5) x = -30*ln(0.5) [after you get the answer, remember it's in months]