alternatives from one age () to the next () are living and dying, the relationship between these two probabilities is: These symbols may also be extended to multiple years, by inserting the number of years at the bottom left of the basic symbol. shows the number of people who die between age and age . is the probability of death between the ages of and age . is the probability that a life age will survive to age . Another statistic that can be obtained from a life table is life expectancy. is the curtate expectation of life for a person alive at age . This is the expected number of complete years remaining to live (you may think of it as the expected number of birthdays that the person will celebrate). A life table generally shows the number of people alive at integral ages. If we need information regarding a fraction of a year, we must make assumptions with respect to the table, if not already implied by a mathematical formula underlying the table. A common assumption is that of a Uniform Distribution of Deaths (UDD) at each year of age. Under this assumption, is a linear interpolation between and . i.e. Annuities The basic symbol for the present value of an annuity is . The following notation can then be added: Notation to the top-right indicates the frequency of payment (i.e., the number of annuity payments that will be made during each year). A lack of such notation means that payments are made annually. Notation to the bottom-right indicates the age of the person when the annuity starts and the period for which an annuity is paid. Notation directly above the basic symbol indicates when payments are made. Two dots indicates an annuity whose payments are made at the beginning of each year (an "annuity-due"); a horizontal line above the symbol indicates an annuity payable continuously (a "continuous annuity"); no mark above the basic symbol indicates an annuity whose payments are made at the end of each year (an "annuity-immediate"). If the payments to be made under an annuity are independent of any life event, it is known as an annuity-certain. Otherwise, in particular if payments end upon the beneficiary's death, it is called a life annuity. (read a-angle-n at i) represents the present value of an annuity-immediate, which is a series of unit payments at the end