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Hazard Rate of a System - YouTube
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hello again um we have uh defined a
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reliability for a system now we want to
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define the hazard and hazard rate
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another rate is stemmed from the
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conditional density
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so the conditional density of x of x is
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a function of x and t obviously x shall
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be bigger than
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t
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the hazard rate is just defined if you
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if you replace both x and t by t
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we
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define it as hazardous
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so uh
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is that the first argument which is not
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uh
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t if you both put so hazard rate means
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just the
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um value of the pdf around t um that
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in in the vicinity of t what is the
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the density of having a failure or is
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called rate of hazard
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so uh because this guy was f x of x
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divided by r of
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rx of t
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so if you replace these by t you get
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this one and this is complementary pdf
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so as i rate is just
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the ratio of the pdf divided by the
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complementary pdf
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you can also see that if you calculate d
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to dt
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of lon of uh
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or x of t
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uh you this is equal to this one because
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this is equal to derivative of uh
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or is equal to minus this one because rx
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of t is 1 minus fx of t
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okay so derivative of logarithm is equal
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to this goes to the denominator and
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derivative of 1 minus f x of t so which
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is going to give you the pdf so the
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hazard rate is
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is that
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you plot the
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and the the reliability reliability
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itself is a
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decreasing function or x of t if you
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plot it in a log scale okay
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um
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minus lon of that
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so
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at this point uh r of t is one
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so the its lawn is zero
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okay and because this function is
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decreasing the lawn should be decreasing
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but because we have a minus here so this
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is going to go up
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so this is minus lon of rx of t so the
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hazard rate is the slope of the
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reliability
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so it says that um so if hazard at
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certain point in time is large so the
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rate of incidence around that point is
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going to be
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very big but if it says constant that
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means that it's
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so the value of beta dt if you multiply
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by dt is the probability that a system
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functioning at time t will fail in the
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interval of t plus dt this is provided
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that dt is very small enough so this is
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an approximation so it is this
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approximation is exact if if dt is very
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small
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but it's a good approximation is dt is
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small enough
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so
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from this expression you can you can you
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can see that this is equal to
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minus derivative of rx of t divided by x
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which is uh minus derivative of the log
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as an example of has arrayed for a
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exponential pdf and the cdf
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1 minus e to the power minus ct
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ut because if you integrate it you get
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this one you will get quickly the hazard
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rate as follows
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so um so here you get
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minus um
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c e to the power minus
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c t
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and then here you get uh e to the power
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minus c t
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and then it will become
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a minus c to become equal to c
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so uh so quickly you you see that
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exponential distribution the hazard is
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constant and this is the only
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distribution which has a constant hazard
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rate that means that as time goes by as
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long as you don't have a failure period
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to the current time
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uh
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the the
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the distribution the shape of
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distribution is unchanged and therefore
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uh the hazard rate is not changed
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so uh this system has a constant hazard
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rate
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um
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the hazard rate uh
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is calculated from in this expression
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you see that
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rx of
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t is equal to 1 minus fx of t
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so using the distribution you can
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calculate the hazard rate
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but we see that the
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the distribution also can be calculated
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using a hazard rate
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always a hazardous
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the negative is positive
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the hazard rate is defined by
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uh by this expression
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now we wonder if we can calculate
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the the distribution pdf or cdf
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provided that the hazard is given to you
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so we can inversely calculate this one
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using the the hazard rate
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so we use the same equations we have
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and obviously this guy is the derivative
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of lon of
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rx of t
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so if this guy is given i could say that
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minus lon of r x of t
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is equal to integral of beta
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x of
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let's say x
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dx
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and x
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varies from zero to time t okay
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and now i could multiply both sides by
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minus sign
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and i say e to the power of both sides e
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to the power of lon becomes this
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function so you get e x b of t so i have
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this formula using which i could
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calculate
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the reliability which is equal to 1
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minus
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f x of t so therefore from here i could
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calculate the cdf and then from there i
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can calculate the pdf
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so um so from here you can you can have
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this one quickly and then see that or x
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of t is equal to e to the power of
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integral of the
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um
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oh this is a t here this should be t
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this should be x because here the dummy
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variable is where x
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so um
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from here if i calculate minus
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derivative of these i get
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this expression so it's minus d2
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dt and then replace t by x
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obviously i get the pdf so this is a
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general formula that you can calculate
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the pdf
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using the
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hazard rate
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so um
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so a system is called memoryless if it's
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hazard is constant
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so this is another definition for a
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memoryless system
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that is the probability that it fails in
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an interval of tx
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assuming that it is functioning at time
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t depends only on the length of these
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intervals so this is another definition
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for
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a memoryless system so a memoryless
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system
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has an exponential uh pdf
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so if you summarize this definition is
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you say that probability that x fails in
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the interval of t and x provided that
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this function at time t
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is a function of the time difference so
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you say that this is a memoryless now
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using this definition you can prove that
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that
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f x of x
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must be equal to c e to the power minus
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c
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x u x there is no other way
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this is for the case of continuous type
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random r but if you have discrete type
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this could shall be you can prove it it
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shall be a geometric distribution
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