The probability distribution of V, the weekly number of accidents at a certain intersection, is given by g(0) = 0.40, g(1) = 0.30, g(2) = 0.20, and g(3) = 0.10. Construct the distribution function of V and draw its graph.
Find the probability that there will be at least two accidents in any one week, using
(a) the original probabilities;
(b) the values of the distribution function.
Solution
The distribution function of $V$ is
$$ \begin{aligned} G(V) &= P(V\leq v) \end{aligned} $$
The distribution function of $V$ is
$$ \begin{aligned} G(v)=\left\{ \begin{array}{ll} 0.40, & \hbox{$v\leq 0$;} \\ 0.40+0.30, & \hbox{$0< v\leq 1$;}\\ 0.40+0.30+0.20, & \hbox{$1< v\leq 2$;}\\ 0.40+0.30+0.20+0.10, & \hbox{$2< v\leq 3$;}\\ \end{array} \right. \end{aligned} $$
Thus the distribution function of $V$ is
$$ \begin{aligned} G(v)=\left\{ \begin{array}{ll} 0.40, & \hbox{$v\leq 0$;} \\ 0.70, & \hbox{$0< v\leq 1$;}\\ 0.90, & \hbox{$1< v\leq 2$;}\\ 1.00, & \hbox{$2< v\leq 3$;}\\ \end{array} \right. \end{aligned} $$
(a) The probability that there will be at least two accidents in any one week, using the original probabilities is
$$ \begin{aligned} P(V\geq 2) &= g(2)+g(3)\\ &= 0.20+0.10\\ &=0.30 \end{aligned} $$
(b) The probability that there will be at least two accidents in any one week, using the value of the distribution function is
$$ \begin{aligned} P(V\geq 2) &= 1- P(V < 2)\\ &= 1- P(V\leq 1)\\ &= 1-G(1)\\ &=1- 0.70\\ &=0.30 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
