The article probabilistic fatigue evaluation of riveted...

The article "Probabilistic Fatigue Evaluation of Riveted Railway Bridges" (J. of Bridge Eng., 2008: 237-244) suggested the exponential distribution with a mean value of 6 MPa as a model for the distribution of stress range in certain bridge connections. Let's assume that this is, in fact, the true model. Find: a) the probability that the stress range is at most 10 MPa, b) between 5 and 10 MPa, c) the median (50th percentile).

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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