The amount of time that a union stays on strike is judged to follow an exponential distribution with a mean of 10 days.
a. Find the probability that a strike lasts less than one day.
b. Find the probability that a strike lasts less than six days.
c. Find the probability that a strike lasts between six and seven days.
d. Find the conditional probability that a strike lasts less than seven days, given that it already has lasted six days. Compare your answer to part a.
Solution
Let $X$ be the time that a union stays on strike . $X\sim \exp(10)$.
The pdf of $X$ is
$$ \begin{aligned} f(x) & = \frac{1}{10}e^{-x/10}, x>0 \end{aligned} $$
The distribution function of $X$ is
$$ \begin{aligned} F(x) & = 1-e^{-x/10} \end{aligned} $$
a. The probability that a strike lasts less than one day is
$$ \begin{aligned} P(X < 1) & = F(1)\\ &= (1 - e^{-1/10})\\ & = 1-e^{-1/10} \\ &= 1-0.9048\\ &= 0.0952 \end{aligned} $$
b. The probability that a strike lasts less than six day is
$$ \begin{aligned} P(X < 6) & = F(6)\\ &= (1 - e^{-6/10})\\ & = 1-e^{-6/10} \\ &= 1-0.5488\\ &= 0.4512 \end{aligned} $$
c. The probability that a strike lasts between six and seven days is
$$ \begin{aligned} P(6 < X < 7) & = F(7)-F(6)\\ &= (1 - e^{-7/10})-(1 - e^{-6/10})\\ & = e^{-6/10} -e^{-7/10}\\ &= 0.5488-0.4966\\ &= 0.0522 \end{aligned} $$
d. The conditional probability that a strike lasts less than seven days, given that it already has lasted six days is
$$ \begin{aligned} P(X < 7|X < 6) &=P(X < 6+1|X < 6)\\ & = P(X < 1)\\ &= 0.9048 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
