# Solved:Let $T$ be the survival time and $T$ follows an exponential distribution.

#### ByDr. Raju Chaudhari

Jul 12, 2020

Let $T$ be the survival time and $T$ follows an exponential distribution. Derive the expression for survival function of $T$. Also obtain expression for force of mortality.

#### Solution

Let $T$ be the survival time. The simplest parametric model for survival data is the exponential distribution with density function

 \begin{aligned} f(t) = \lambda e^{-\lambda t}, t> 0, \lambda >0. \end{aligned}

Distribution function of exponential dist.

The distribution function of $T$ is

 \begin{align} F(t) & = P(T Thus the survival function is  \begin{aligned} S(t) = 1-F(t) = e^{-\lambda t}. \end{aligned} $$The force of mortality is given by$$ \begin{aligned} \mu(t) & = \frac{f(t)}{1-F(t)}=-\frac{S^\prime(t)}{S(t)}\\ &= \frac{\lambda e^{-\lambda t}}{1-\big(1-e^{-\lambda t}\big)}\\ &=\frac{\lambda e^{-\lambda t}}{e^{-\lambda t}}\\ &= \lambda . \end{aligned} $$OR The force of mortality is given by$$ \begin{aligned} \mu(t) & = -\frac{S^\prime(t)}{S(t)}\\ &= -\frac{-\lambda e^{-\lambda t}}{e^{-\lambda t}}\\ &= \lambda . \end{aligned} $$Further Reading Statistics Descriptive Statistics Probability Theory Probability Distribution Hypothesis Testing Confidence interval Sample size determination Non-parametric Tests Correlation Regression Statistics Calculators function pinIt() { var e = document.createElement('script'); e.setAttribute('type','text/javascript'); e.setAttribute('charset','UTF-8'); e.setAttribute('src','https://assets.pinterest.com/js/pinmarklet.js?r='+Math.random()*99999999); document.body.appendChild(e); }     By Dr. Raju Chaudhari Related Post Estimation Statistics Solved: A survey of 120 female freshman showed that 18 did not wish to work after marriage. 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