- #1

Eclair_de_XII

- 1,083

- 91

## Homework Statement

"Suppose that the accident rate for one workplace ##A## is ##k## times the rate of another workplace ##B##. In other words, ##\lambda_A(t)=k⋅\lambda_B(t)##. Conclude that the probability of no accidents in workplace ##A## is the probability of no accidents in workplace ##B## to the ##k##-th power.

In other words, if ##\lambda_A(t)=k⋅\lambda_B(t)##, then ##P(X^c|A)=P(X^c|B)^k## where ##X## denotes the event of an accident."

## Homework Equations

Hazard function: ##\lambda(t)=\frac{F'(t)}{1-F(t)}##

-denotes the probability of an accident happening on ##t+\delta t## given no accidents have happened by ##t##

## The Attempt at a Solution

I'm having trouble interpreting this function, honestly. I mean, what do I get if I integrate ##\lambda(t)##? Is it not just the probability of an accident happening between some initial time ##t_0## and some final time ##t_f## for either workplace? What I got, through integration, was:

(Assuming ##\lambda## is constant)

##\lambda_B(t)=\frac{F'(t)}{1-F(t)}##

##\lambda_Bt=\frac{F'(t)}{1-F(t)}dt##

##\lambda_Bt=\int_{t_0}^{t_f} \frac{F'(t)}{1-F(t)}dt=P(X|B)##

##P(X|B)=-\int_{t_0}^{t_f} \frac{F'(t)}{F(t)-1}dt=-\int_{t_0}^{t_f} \frac{F'(t)}{F(t)-1}dt=-ln|F(t)-1|=ln|F(t_0)-1|-ln|F(t_f)-1|=ln|\frac{F(t_0)-1}{F(t_f)-1}|##

Then what I get for ##P(X|A)## is:

##P(X|A)=-k⋅\int_{t_0}^{t_f} \frac{F'(t)}{F(t)-1}dt=-k⋅ln|F(t)-1|=k⋅[ln|F(t_0)-1|-ln|F(t_f)-1|]=k⋅ln|\frac{F(t_0)-1}{F(t_f)-1}|=ln|\frac{F(t_0)-1}{F(t_f)-1}|^k=P(X|B)^k##

I'm confused on how ##\lambda(t)## works... Can anyone tell me how to interpret its integral? Can anyone tell me if I integrated the wrong function?

Last edited: