Find the cumulative distribution function for each continuous probability distribution:
f \left(x \right)=\begin{cases} \dfrac{x^{2}}{9}, & 0 \leq x \leq 3 \\ 0, & \text{otherwise} \end{cases}
f \left(x \right)=\begin{cases} \dfrac{e^{x}}{e^{4} - 1}, & 0 \leq x \leq 4 \\ 0, & \text{otherwise} \end{cases}
f \left(x \right)=\begin{cases} \dfrac{3 x \left(8 - x\right)}{135}, & \text{for }x \text{ in }\left[2, 5\right] \\ 0, & \text{otherwise} \end{cases}
Consider the probability density function f \left(x \right)=\begin{cases} \dfrac{5x^4}{7775}, & \text{for }1 \leq x \leq 6 \\ 0, & \text{otherwise} \end{cases}.
Find the cumulative distribution function.
Calculate P \left( X \leq 2 \right).
Calculate P \left( X \lt 5 \right).
Calculate P \left( 2 \leq X \leq 4 \right).
A continuous probability distribution is given by f \left(x \right)=\begin{cases} \dfrac{4x^3}{2320}, & \text{for }x \text{ in }\left[3, 7\right] \\ 0, & \text{otherwise} \end{cases}.
Find the cumulative distribution function.
Calculate P \left( X \leq 6 \right).
Calculate P \left( X \geq 5 \right).
Calculate P \left( 4 \leq X \lt 6 \right).
A continuous probability distribution is defined by f \left(x \right)=\begin{cases} \dfrac{2e^{2x}}{e^{10}-1}, & \text{for }x \text{ in }\left[0, 5\right] \\ 0, & \text{otherwise} \end{cases}.
Find the cumulative distribution function.
Calculate P\left(X\leq 4 \right) correct to two decimal places.
Calculate P \left(X\gt 3 \right) correct to two decimal places.
Calculate P \left(2\leq X\leq 4 \right) correct to two decimal places.
A continuous probability distribution is defined by f \left( x \right)=\begin{cases} ax^3, & \text{for }x \text{ in }\left[0, 9\right] \\ 0, & \text{otherwise} \end{cases}.
Evaluate a.
Find the cumulative distribution function.
Calculate P \left(X\leq 5 \right).
Calculate P \left(X\leq 4 \right).
Calculate P \left(X\gt 8 \right).
Calculate P \left(X\geq 3 \right).
Calculate P \left(2\leq X\leq 6 \right).
A continuous probability distribution is defined by f \left(x \right)=\begin{cases} \dfrac{a}{x}, & \text{for }x \text{ in }\left[1, 6\right] \\ 0, & \text{otherwise} \end{cases}.
Find the exact value of a.
Find the cumulative distribution function.
Calculate P \left(X\leq 2 \right) correct to two decimal places.
Calculate P \left(X\gt 5\right) correct to two decimal places.
Calculate P \left(2\leq X\leq 5\right) correct to two decimal places.
Consider the probability density function f\left(x \right)=\begin{cases} \cos x, & \text{for }x \text{ in }\left[\dfrac{3\pi}{2},2\pi\right] \\ 0, & \text{otherwise} \end{cases}.
Find the cumulative distribution function.
Calculate P\left(X\leq \dfrac{5\pi}{3}\right).
Calculate P\left(X\geq \dfrac{7\pi}{4}\right).
Consider the probability density function f\left(x \right)=\begin{cases} c, & \text{for }0 \leq x \leq 5 \\ 2c, & \text{for }5 \lt x \leq 10 \\ 0, & \text{otherwise} \end{cases}.
Sketch the probability density function.
Find the value of c.
Find the cumulative distribution function.
Calculate P \left(1\lt X\lt 7\right).
Find the mode of each continuous probability density function:
\\ f\left(x \right)=\begin{cases} \dfrac{3 \left(9+8x-x^2\right)}{434}, & \text{for }x \text{ in } \left[0,7\right]\\ 0, & \text{otherwise} \end{cases}
\\ f\left(x \right)=\begin{cases} \dfrac{4e^{4x}}{e^8 \left(e^{16}-1\right)}, & \text{for }2 \leq x \leq 6\\ 0, & \text{otherwise} \end{cases}
Consider the probability density functionf\left(x \right)=\begin{cases} -\dfrac{3}{22}\left(x^2-6x+5 \right), & \text{for }x \text{ in }\left[2,4\right]\\ 0, & \text{otherwise} \end{cases}.
Calculate the mode.
Calculate P \left( X \leq a \right), where a is the mode.
Find the cumulative distribution function.
The times that athletes took to finish a race varied between 3 and 7 minutes and are represented by the following probability density function:f\left(x \right)=\begin{cases} \dfrac{1}{116} \left(x^3-9x^2+24x+1\right), & \text{for }x \text{ in } \left[3,7\right]\\ 0, & \text{otherwise} \end{cases}
Find the cumulative distribution function.
Calculate the probability that an athlete finishes the race in less than 5 minutes.
Calculate the probability that an athlete finishes the race in 4 minutes or more.
Find the probability that the time it takes an athlete to finish the race is between 4 and 5 minutes.
Find the value of x in \left(3, 7\right) for which f' \left( x \right) = 0.
Complete the following table of values:
| x | 3 | 3.5 | 4 | 5 | 7 |
|---|---|---|---|---|---|
| f(x) |
Hence, find the most frequent time in which an athlete would finish the race.
Find the 1st quartile for the probability density function f\left(x \right)=\begin{cases} \dfrac{3x^2}{973}, & \text{for }x \text{ in } \left[3,10 \right]\\ 0, & \text{otherwise} \end{cases}.
Round your answer to two decimal places.
Find the 2nd decile for the probability density function f\left(x \right)=\begin{cases} \dfrac{x^3}{324}, & \text{for }0 \leq x\leq 6\\ 0, & \text{otherwise} \end{cases}.
Round your answer to two decimal places.
Find the 77th percentile for the probability density function f\left(x \right)=\begin{cases} \dfrac{5x^4}{3124}, & \text{for }1 \leq x\leq 5\\ 0, & \text{otherwise} \end{cases}.
Round your answer to two decimal places.
Consider the probability density function f\left(x \right)=\begin{cases} \dfrac{3x^2}{512}, & \text{for }x \text{ in } \left[0,8\right]\\ 0, & \text{otherwise} \end{cases}.
Calculate the following to two decimal places:
Median
35\text{th} percentile
Consider the probability density function f\left(x \right)=\begin{cases} \dfrac{x^2}{168}, & \text{for }2\leq x \leq 8\\ 0, & \text{otherwise} \end{cases}.
Calculate the following to two decimal places:
Median
3rd quartile
67th percentile
8th decile
Consider the probability density function f\left(x \right)=\begin{cases} \dfrac{x^2}{576}, & \text{for }0\leq x \leq 12\\ 0, & \text{otherwise} \end{cases}.
Find the 20th percentile to two decimal places.
Consider the probability density function: f\left(x \right)=\begin{cases} \dfrac{x^3}{1020}, & \text{for }2\leq x \leq 8\\ 0, & \text{otherwise} \end{cases}.
Find the cumulative distribution function.
Find P \left(X \leq 5 \right).
Find P \left(3 \leq X \leq 7 \right).
Find the 9th decile correct to two decimal places.
Find the 23rd percentile correct to two decimal places.
Consider the function f \left( x \right)=\begin{cases} ce^{-x}, & \text{for }0\leq x \leq 1\\ 0, & \text{otherwise} \end{cases}.
Find c, given that f \left( x \right) is a probability density function.
Find the cumulative distribution function.
Find the 1st quartile correct to two decimal places.
Find the 2nd quartile correct to two decimal places.
Find the 3rd quartile correct to two decimal places.
Yan does not know the times when trains leave Lakeside Station, but he does know they leave precisely every fifteen minutes.
Find the probability density function of his waiting time if he arrives at the station at a random time.
Find the cumulative distribution function of his waiting time if he arrives at the station at a random time.
Find the 45th percentile.
Find the probability that he will wait between 5 and 10 minutes.
Buses go into the city every 15 minutes, so Susana doesn't bother looking at the timetable before catching the bus.
Construct a probability density function f \left( t \right) where t is the number of minutes for the next bus when she reaches the stop.
Construct a cumulative distribution function F \left( t \right) where t is the number of minutes for the next bus when she reaches the stop.
Find the median waiting time.
Find the value of a, the 45th percentile.
Calculate the probability that he will wait between 5 and 7 minutes.