Give some evualation as a consequence of UDD to tpyµy(t).

 The expression you provided is derived from the ultimate probability of survival to age

$�$ for a life aged $�$, denoted as $�{�}_{�}$, where $�$ is an integer and $0<�<1$.

The given expression is:

$-\frac{�}{��}�{�}_{�}=-\frac{�}{��}(1-�{�}_{�})={�}_{�}$

Here's the evaluation of this consequence of the ultimate probability of death distribution (UDD) to $�{�}_{�}$:

1. Interpretation:

   • $�{�}_{�}$ represents the probability that a life aged $�$ will survive to age $�$.
   • ${�}_{�}$ represents the ultimate probability of dying between ages $�$ and $�+1$, which is essentially the complement of $�{�}_{�}$, i.e., ${�}_{�}=1-�{�}_{�}$.
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2. Derivation:

   • Taking the derivative of $�{�}_{�}$ with respect to $�$ involves differentiating the function $1-�{�}_{�}$ as it's an alternative representation of $�{�}_{�}$. This is because ${�}_{�}=1-�{�}_{�}$.
   • The derivative $\frac{�}{��}(1-�{�}_{�})$ simplifies to ${�}_{�}$ because ${�}_{�}$ is constant with respect to $�$.
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3. Consequence:

   • This consequence demonstrates a direct relationship between the derivative of the ultimate probability of survival $�{�}_{�}$ and the ultimate probability of death ${�}_{�}$.
   • Essentially, it shows that the rate of change of the probability of survival $�{�}_{�}$ with respect to age $�$ is equal to the ultimate probability of death ${�}_{�}$ at that age.
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4. Implication:

   • This relationship is fundamental in actuarial science and insurance mathematics. It's used extensively in calculating premiums, reserves, and other actuarial quantities.
   • It simplifies the calculation of survival probabilities and mortality rates, making it easier to model and analyze life insurance and annuity products.

In summary, the consequence of the UDD to $�{�}_{�}$ simplifies the calculation of survival probabilities by providing a direct relationship between the derivative of the ultimate probability of survival and the ultimate probability of death. This relationship is crucial in actuarial science for various calculations and analyses.