The lifetimes in years for a particular brand of cathode ray tube are exponentially distributed with a mean of 5 years. What percent of the tubes have lifetimes between 5 and 8 years? Draw a graph of the pdf and shade the area which represents the probability of the event $5 < X < 8$, where $X$ represents the lifetimes.
Solution
Let $X$ be lifetimes in years for a particular brand of cathode ray tube. $X\sim \exp(5)$.
The pdf of $X$ is
$$ \begin{aligned} f(x) & = \frac{1}{5}e^{-x/5}, x>0 \end{aligned} $$
The distribution function of $X$ is
$$ \begin{aligned} F(x) & = P(X\leq x)\\ &=1-e^{-x/5} \end{aligned} $$
The probability that the tubes have lifetimes between 5 and 8 years is
$$ \begin{aligned} P(5 < X < 8) & = F(8)-F(5)\\ &= (1 - e^{-8/5})-(1 - e^{-5/5})\\ & = e^{-5/5} -e^{-8/5}\\ &= 0.3679-0.2019\\ &= 0.166 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
