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What is the probability that a process is alive at time T? What is the right unit for T?
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I have a continuous process. At any time $t$, the probability that it fails is given by the continuous but not necessarily monotonic function $p_fail(t)$.
Based on this, I have two questions:
- What is the probability that the process does not fail within the interval $[0,T]$?
- How do I get the units right / how can I make the probability independent of the unit, which the time is given in?
This is what I tried so far:
Question 1
A) Product Integral
Consider a continuous event tree, that splits into two branches at each layer: failed + not failed. A failed node does not expand further. Applying the multiplication rule to the not failed chain gives the probability that the process is still alive at time $T$:
$$ beginalign
P_alive(T) &= prod_t=0^T(1 - p_fail(t)) \
&= exp(int_t=0^T ln(1-p_fail(t)) dt)
endalign $$
B) Reliability Approach
Consider the function $p_fail(t)$ to give the failure rate at time $t$. The probability that the process is alive at time $T$ is given by the reliability function:
$$ beginalign
R(T) = exp(-int_t=0^T p_fail(t) dt)
endalign $$
For both approaches, results are very similar in my case (difference is between $10^-10$ and $10^-15$)
Question 2
The results of both approaches obviously depend on the unit the time is given in.
So my second problem is to get the units right.
Consider the simple function $p_1,fail(t [s]) = frac0.013600s cdot t$, for $t$ in seconds. This (in my opinion) is equal to $p_2,fail(t [h]) = frac0.011h cdot t$, for $t$ in hours.
With $T = 1h = 3600s$ the integration gives different results, depending on the unit of time:
beginarray c
hline & int_t=0^T ln(1-p(t)) dt & int_t=0^T p(t) dt \
hline p_1 & -18.00 & 18.00 \
hline p_2 & -0.005 & 0.005 \
hline
endarray
Of course, this affects the result for $P_alive(T)$. Maybe this is an indicator, that both approaches, A and B, are wrong?
probability unit-of-measure reliability
asked Aug 6 at 8:49
Stanley F.
1064
add a comment |Â
up vote
0
down vote
favorite
I have a continuous process. At any time $t$, the probability that it fails is given by the continuous but not necessarily monotonic function $p_fail(t)$.
Based on this, I have two questions:
- What is the probability that the process does not fail within the interval $[0,T]$?
- How do I get the units right / how can I make the probability independent of the unit, which the time is given in?
This is what I tried so far:
Question 1
A) Product Integral
Consider a continuous event tree, that splits into two branches at each layer: failed + not failed. A failed node does not expand further. Applying the multiplication rule to the not failed chain gives the probability that the process is still alive at time $T$:
$$ beginalign
P_alive(T) &= prod_t=0^T(1 - p_fail(t)) \
&= exp(int_t=0^T ln(1-p_fail(t)) dt)
endalign $$
B) Reliability Approach
Consider the function $p_fail(t)$ to give the failure rate at time $t$. The probability that the process is alive at time $T$ is given by the reliability function:
$$ beginalign
R(T) = exp(-int_t=0^T p_fail(t) dt)
endalign $$
For both approaches, results are very similar in my case (difference is between $10^-10$ and $10^-15$)
Question 2
The results of both approaches obviously depend on the unit the time is given in.
So my second problem is to get the units right.
Consider the simple function $p_1,fail(t [s]) = frac0.013600s cdot t$, for $t$ in seconds. This (in my opinion) is equal to $p_2,fail(t [h]) = frac0.011h cdot t$, for $t$ in hours.
With $T = 1h = 3600s$ the integration gives different results, depending on the unit of time:
beginarray c
hline & int_t=0^T ln(1-p(t)) dt & int_t=0^T p(t) dt \
hline p_1 & -18.00 & 18.00 \
hline p_2 & -0.005 & 0.005 \
hline
endarray
Of course, this affects the result for $P_alive(T)$. Maybe this is an indicator, that both approaches, A and B, are wrong?
probability unit-of-measure reliability
asked Aug 6 at 8:49
Stanley F.
1064
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a continuous process. At any time $t$, the probability that it fails is given by the continuous but not necessarily monotonic function $p_fail(t)$.
Based on this, I have two questions:
- What is the probability that the process does not fail within the interval $[0,T]$?
- How do I get the units right / how can I make the probability independent of the unit, which the time is given in?
This is what I tried so far:
Question 1
A) Product Integral
Consider a continuous event tree, that splits into two branches at each layer: failed + not failed. A failed node does not expand further. Applying the multiplication rule to the not failed chain gives the probability that the process is still alive at time $T$:
$$ beginalign
P_alive(T) &= prod_t=0^T(1 - p_fail(t)) \
&= exp(int_t=0^T ln(1-p_fail(t)) dt)
endalign $$
B) Reliability Approach
Consider the function $p_fail(t)$ to give the failure rate at time $t$. The probability that the process is alive at time $T$ is given by the reliability function:
$$ beginalign
R(T) = exp(-int_t=0^T p_fail(t) dt)
endalign $$
For both approaches, results are very similar in my case (difference is between $10^-10$ and $10^-15$)
Question 2
The results of both approaches obviously depend on the unit the time is given in.
So my second problem is to get the units right.
Consider the simple function $p_1,fail(t [s]) = frac0.013600s cdot t$, for $t$ in seconds. This (in my opinion) is equal to $p_2,fail(t [h]) = frac0.011h cdot t$, for $t$ in hours.
With $T = 1h = 3600s$ the integration gives different results, depending on the unit of time:
beginarray c
hline & int_t=0^T ln(1-p(t)) dt & int_t=0^T p(t) dt \
hline p_1 & -18.00 & 18.00 \
hline p_2 & -0.005 & 0.005 \
hline
endarray
Of course, this affects the result for $P_alive(T)$. Maybe this is an indicator, that both approaches, A and B, are wrong?
probability unit-of-measure reliability
asked Aug 6 at 8:49
Stanley F.
1064
I have a continuous process. At any time $t$, the probability that it fails is given by the continuous but not necessarily monotonic function $p_fail(t)$.
Based on this, I have two questions:
- What is the probability that the process does not fail within the interval $[0,T]$?
- How do I get the units right / how can I make the probability independent of the unit, which the time is given in?
This is what I tried so far:
Question 1
A) Product Integral
Consider a continuous event tree, that splits into two branches at each layer: failed + not failed. A failed node does not expand further. Applying the multiplication rule to the not failed chain gives the probability that the process is still alive at time $T$:
$$ beginalign
P_alive(T) &= prod_t=0^T(1 - p_fail(t)) \
&= exp(int_t=0^T ln(1-p_fail(t)) dt)
endalign $$
B) Reliability Approach
Consider the function $p_fail(t)$ to give the failure rate at time $t$. The probability that the process is alive at time $T$ is given by the reliability function:
$$ beginalign
R(T) = exp(-int_t=0^T p_fail(t) dt)
endalign $$
For both approaches, results are very similar in my case (difference is between $10^-10$ and $10^-15$)
Question 2
The results of both approaches obviously depend on the unit the time is given in.
So my second problem is to get the units right.
Consider the simple function $p_1,fail(t [s]) = frac0.013600s cdot t$, for $t$ in seconds. This (in my opinion) is equal to $p_2,fail(t [h]) = frac0.011h cdot t$, for $t$ in hours.
With $T = 1h = 3600s$ the integration gives different results, depending on the unit of time:
beginarray c
hline & int_t=0^T ln(1-p(t)) dt & int_t=0^T p(t) dt \
hline p_1 & -18.00 & 18.00 \
hline p_2 & -0.005 & 0.005 \
hline
endarray
Of course, this affects the result for $P_alive(T)$. Maybe this is an indicator, that both approaches, A and B, are wrong?
probability unit-of-measure reliability
asked Aug 6 at 8:49
Stanley F.
1064
asked Aug 6 at 8:49
Stanley F.
1064
asked Aug 6 at 8:49
Stanley F.
1064
asked Aug 6 at 8:49
Stanley F.
1064
1064
add a comment |Â
add a comment |Â
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