Sensor measured strain x in microstrains on a bridge during...

Name: sensor measured strain x in microstrains on a bridge during the passage of nine goods trains train 1 2 3 4 5 6 7 8 9 strain x 15 86 03 37 68 42 17 78 90 for these data x 436 and x2 29860 ass 37176
Uploaded: 2023-01-09T05:19:43-08:00
Duration: 7 min 56 s
Channel: Sri K
Description: sensor measured strain x in microstrains on a bridge during the passage of nine goods trains train 1 2 3 4 5 6 7 8 9 strain x 15 86 03 37 68 42 17 78 90 for these data x 436 and x2 29860 ass 37176

sensor measured strain x (in microstrains) on a bridge during the passage of nine goods trains: Train: 1 2 3 4 5 6 7 8 9 Strain x: 1.5 8.6 0.3 3.7 6.8 4.2 1.7 7.8 9.0 For these data ∑ x = 43.6 and ∑ x2 = 298.60 Assume that strain on the bridge can be considered to be a Nor- mal random variable with population mean μ = 5 and population standard deviation σ = 2.5. (i) What is the probability that the next goods train will lead to a strain in excess of 9 microstrains? (ii) What is the probability distribution of the mean strain imposed by n = 12 goods trains? (iii) What is the probability distribution of the number of trains, out of 12 crossing the bridge, that lead to strains in excess of 9 microsrains?

Transcript

00:01 In this problem, the stress range has given in the question, stress range is random variable and it follows the exponential distribution with the parameter lambda is equal to 1 upon 6, which is the density function, density function.

00:46 And therefore we will take here a function of x for 1 by 6.

00:52 6, e to the power of minus 1 by 6 x when x is greater than or equal to 0 and it will be 0 otherwise.

01:07 So now in the solution of first part of the question part a, the proportions of stress range are at least to mpa mega pepals which is in equal to integration of 2 to infinity for 1 by 6 e to the power of minus 1 upon 6 x d x and from here we will get our value which is equal to a to the power of minus 2 by 6 when we read this exponential value this is equal to 0 .7165 so hence the proportion proportion of of stress at the least 2 megapixel of stress range are leased at 2 mphe.

02:36 Megapascal is 0 .478 and again we will check the proportion of stress range are at it most 7 mp a whereas integration of minus infinity to 7 for 1 by 6 e to the power of minus 1 upon 6 x d x so this will be calculated further as opening the integration value minus infinity to 0 d x plus 0 to 7 1 upon 6 e to 1 1 upon 6 e to 1 1 1 1 6 8 x t x so our first tongue will become 1 so 1 minus e to the bar of minus 7 by 6 and its exponential value is 0 .6886 hence the proportion ranges now 0 .6886 at 7 mp a so the proportion of stress range is in between proportion proportion proportion stress range ranges are in between 5 and 10 megabascals which is calculated as integration of 5 to 10 for 1 by 6 e to the power of minus 1 upon 6 d x and calculation for this part will be 8 to the power of minus 5 by 6 minus e to the power of minus 10 by 6 further it is equal to 0 .24571 so our answer for part a is hence the proportion of stress range are between 5 to 10 is 0 .2457 is 0 .2 4757 so we are done with the first part of the question.

05:23 Here we have calculated all the three stress ranges proportion.

05:28 As we have calculated first one here, this is our first solution...

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