% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/multilife_ch12.R
\name{multilife_survival_probabilities}
\alias{multilife_survival_probabilities}
\alias{tpxy}
\alias{tqxy}
\alias{tpxybar}
\alias{tqxybar}
\title{Multi-life survival and failure probabilities}
\usage{
tpxy(x, y, t, tbl = NULL, model = NULL, ...)

tqxy(x, y, t, tbl = NULL, model = NULL, ...)

tpxybar(x, y, t, tbl = NULL, model = NULL, ...)

tqxybar(x, y, t, tbl = NULL, model = NULL, ...)
}
\arguments{
\item{x}{Age of the first life. May be scalar or vector.}

\item{y}{Age of the second life. May be scalar or vector.}

\item{t}{Nonnegative duration. May be scalar or vector.}

\item{tbl}{Optional life table object.}

\item{model}{Optional parametric survival model.}

\item{...}{Additional parameters passed to the survival model.}
}
\value{
A numeric vector of probabilities.
}
\description{
Computes joint-life and last-survivor survival and failure probabilities
for two independent lives.
}
\details{
\code{tpxy()} computes the joint-life survival probability
\deqn{
{}_tp_{xy}
=
{}_tp_x\,{}_tp_y.
}

\code{tqxy()} computes the joint-life failure probability
\deqn{
{}_tq_{xy}
=
1-{}_tp_{xy}.
}

\code{tpxybar()} computes the last-survivor survival probability
\deqn{
{}_tp_{\overline{xy}}
=
{}_tp_x + {}_tp_y - {}_tp_x\,{}_tp_y.
}

\code{tqxybar()} computes the last-survivor failure probability
\deqn{
{}_tq_{\overline{xy}}
=
1-{}_tp_{\overline{xy}}.
}


All calculations assume the future lifetimes are independent.

Joint-life probabilities require both lives to satisfy the survival
condition, whereas last-survivor probabilities require at least one life
to survive.
}
\examples{
tpxy(
  40, 50,
  t = 10,
  model = "uniform",
  omega = 100
)

tqxy(
  40, 50,
  t = 10,
  model = "uniform",
  omega = 100
)

tpxybar(
  40, 50,
  t = 10,
  model = "uniform",
  omega = 100
)

tqxybar(
  40, 50,
  t = 10,
  model = "uniform",
  omega = 100
)

}
