Actuarial Statistics
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\(S_0(t) = \)
\(S_0(t) = Pr(T_0 > t) = {}_tp_0\)
Probability that (0) dies after t periods (years) => survival function of \(T_0\)
where:
- \({}_tp_0\)represents the probability that a person aged x survives for at least t years (the t-year survival probability).
\(T_0\)
Time-until-death of a new-born baby
\(F_0(t) = \)
\(F_0(t) = Pr(T_0 ≤ t) = 1- S_0(t) = ...?\)
cummulative distribution function
p.d.f.
Probability Density Function: describes the likelihood of a continuous random variable taking on a particular value, it is used to determine the probability that the variable falls within an interval by integrating the p.d.f. over that interval.
Properties of a p.d.f.
- always non-negative
- The total area under the p.d.f. curve equals 1, ensuring that the total probability over all possible values of the random variable sums to 1.
Example of p.d.f.
If \(f_0(t)\) is the p.d.f. of a random variable X, the probability that X falls between two values a and b is given by:
\(P(a≤X≤b)= ∫_{a}^{b} f_x(x)dx\)
For a continuous random variable \(T_0\) with a p.d.f. \(f_{T_0}(t)\), the expected value of \(T_0\) is given by:
.
This formula is equal to ? ITO survival function and curtate life expectancy
.
What means \(µ_t\)?
the force of mortality: measures the instantaneous rate of mortality at a specific age x.
Formula, where:
- \(f(x)\) is the p.d.f. of the age-at-death r.v. X. X. It describes the likelihood of death at a specific age x.
- S(x) is the survival function, which gives the probability that an individual aged x survives beyond age x.
- also: the force of mortality is the negative derivative of the natural logarithm of the survivor function.
Cumulative distribution function (CDF)
\(F_0(t) = Pr(T_0 ≤ T) = 1 - S_0(t) = tq_0 \)
CDF of the age-at-death random variable, representing the probability of dying at age xxx or earlier.
where:
- \(tq_0 \)is the probability that a person aged x will die within the next t years.
\({}_tp_x + tq_x = 1\)
- \({}_tp_x \) represents the probability that a person aged x survives for at least t years (the t-year survival probability).
- \(tq_x \) The probability that a person aged x will die within the next t years.
--> they are complementary probabilities
relation b/w survival function, CDF, P and q
.
Formula? given a specific survival function.
p.d.f. of \(T_0\)
\(µ_t\) formula
relation b/w \(µ_x\) and \({}_xp_0\)
\({}_xp_0\) is the probability that (0) (a person of age 0) dies after x periods (years) and is therefore the same as the survival function \(S_0(t) = Pr(T_0 > t)\)
Beta distribution
The random variable X is called a beta random variable if its p.d.f is given by
ITO E[X]
Erwartungswert und Varianz - Regeln
Erwartungswert und Varianz von Summen
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Cartes-fiches
\(S_0(t) = \)
\(S_0(t) = Pr(T_0 > t) = {}_tp_0\)
Probability that (0) dies after t periods (years) => survival function of \(T_0\)
where:
- \({}_tp_0\)represents the probability that a person aged x survives for at least t years (the t-year survival probability).
\(T_0\)
Time-until-death of a new-born baby
\(F_0(t) = \)
\(F_0(t) = Pr(T_0 ≤ t) = 1- S_0(t) = ...?\)
cummulative distribution function
p.d.f.
Probability Density Function: describes the likelihood of a continuous random variable taking on a particular value, it is used to determine the probability that the variable falls within an interval by integrating the p.d.f. over that interval.
Properties of a p.d.f.
- always non-negative
- The total area under the p.d.f. curve equals 1, ensuring that the total probability over all possible values of the random variable sums to 1.
Example of p.d.f.
If \(f_0(t)\) is the p.d.f. of a random variable X, the probability that X falls between two values a and b is given by:
\(P(a≤X≤b)= ∫_{a}^{b} f_x(x)dx\)
For a continuous random variable \(T_0\) with a p.d.f. \(f_{T_0}(t)\), the expected value of \(T_0\) is given by:
.
This formula is equal to ? ITO survival function and curtate life expectancy
.
What means \(µ_t\)?
the force of mortality: measures the instantaneous rate of mortality at a specific age x.
Formula, where:
- \(f(x)\) is the p.d.f. of the age-at-death r.v. X. X. It describes the likelihood of death at a specific age x.
- S(x) is the survival function, which gives the probability that an individual aged x survives beyond age x.
- also: the force of mortality is the negative derivative of the natural logarithm of the survivor function.
Cumulative distribution function (CDF)
\(F_0(t) = Pr(T_0 ≤ T) = 1 - S_0(t) = tq_0 \)
CDF of the age-at-death random variable, representing the probability of dying at age xxx or earlier.
where:
- \(tq_0 \)is the probability that a person aged x will die within the next t years.
\({}_tp_x + tq_x = 1\)
- \({}_tp_x \) represents the probability that a person aged x survives for at least t years (the t-year survival probability).
- \(tq_x \) The probability that a person aged x will die within the next t years.
--> they are complementary probabilities
relation b/w survival function, CDF, P and q
.
Formula? given a specific survival function.
p.d.f. of \(T_0\)
\(µ_t\) formula
relation b/w \(µ_x\) and \({}_xp_0\)
\({}_xp_0\) is the probability that (0) (a person of age 0) dies after x periods (years) and is therefore the same as the survival function \(S_0(t) = Pr(T_0 > t)\)
Beta distribution
The random variable X is called a beta random variable if its p.d.f is given by
ITO E[X]
Erwartungswert und Varianz - Regeln
Erwartungswert und Varianz von Summen
