Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the \\x-axis.
State the features of p \left( x \right) which are also features of all probability distribution functions.
Calculate P \left(X \leq 54 \right).
Calculate P \left(X \gt 34 \right).
Calculate P \left(44 \lt X \leq 53 \right).
Calculate P \left(X \leq 56 | X \geq 44 \right) .
Consider the probability density function p \left( x \right) = \dfrac{1}{40} for 60 \leq x \leq 100 and p \left( x \right) = 0 otherwise.
Sketch the graph of the probability density function p \left(x \right).
Calculate P \left( X \gt 80 \right).
Calculate P \left(X \leq 65 | X \leq 90 \right).
Calculate P \left(X \leq 70 \cap X \geq 80 \right).
Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the \\ x-axis.
Calculate P \left(X \lt 3 \right).
Calculate P \left(X \gt 3 | X \leq 4 \right).
Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the \\x-axis.
Calculate P \left (\dfrac{11}{2} \leq X \lt \dfrac{17}{2} \right).
Calculate P \left (X \lt \dfrac{17}{2}|X \geq \dfrac{11}{2} \right).
Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the \\ x-axis.
State the probability density function for X.
Calculate P \left(X \lt 2|X \gt 1 \right).
Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the \\ x-axis.
State the probability density function for X.
Calculate P \left(X \geq 4 \right).
Calculate P \left(X \gt 3|X \lt 4 \right).
Consider the semi-elliptical probability density function p \left( x \right) drawn below for a random variable X. The maximum height the function reaches is \dfrac{1}{2 \pi}.
Using the fact that the formula for calculating the area of a semi-ellipse is \text{Area} = \dfrac{\pi a b}{2}, where a the distance the \\ x-intercepts are from the centre of the ellipse, and b is the maximum height the ellipse reaches, calculate the area between p \left( x \right) and the x-axis.
Find the median value for X.
Find the probability that X is less than the median.
Consider the probability density function p \left( x \right) drawn below for a random variable X:
Calculate the area between p \left( x \right) and the x-axis.
Given that p \left( 8 \right) = \dfrac{19}{160}, calculate P(X \gt 8) using geometric reasoning.
Given that p \left( 12 \right) = \dfrac{23}{160}, calculate P(X \lt 12).
Given that p \left( 7 \right) = \dfrac{9}{80} and p \left( 11 \right) = \dfrac{11}{80}, calculate P(X \geq 7 | X \leq 11).
State the probability density function for X.
Hence, calculate the median value (m) of X, correct to two decimal places.
Consider the probability density function:
p\left( x \right) = \begin{cases} \dfrac{-x}{40}+\dfrac{1}{5}, & 0 \leq x \leq 8 \\ \dfrac{x}{160}-\dfrac{1}{20}, & 8 \lt x \leq 16 \\ 0, & \text{otherwise} \end{cases}
Sketch the graph of the probability density function p\left( x \right).
Calculate the area under p \left( x \right).
Explain the reason why p \left( x \right) is a probability density function.
The probability density function of a random variable X is drawn below. Its non-zero values lie in the region 0 \leq x \leq 14.
Calculate the value of k.
Find the equation that defines the probability distribution function of X in the domain 5 \leq x \leq 14.
Calculate P(X \lt 8). Give your answer in exact form.
Hence, calculate P( X \geq 2 | X \leq 8 ). Give your answer in exact form.
The probability density function of a random variable X is shown. Its non-zero values lie in the region 0 \leq x \leq k:
Calculate the value of k.
State the equation that defines the probability distribution function of X in the domain 0 \leq x \leq k.
Calculate P \left(X \lt 0.8 k \right) to two decimal places.
Calculate P \left(X \geq 0.2 k \right) to two decimal places.
Calculate P \left(X \lt 7 | X \gt 2 \right).
The probability density function of a random variable X is defined:
p \left( x \right) = \begin{cases} kx & \quad 0\leq x \leq 7\\ 7k & \quad 7 \lt x \leq 10\\ 0 & \quad \text{otherwise} \end{cases}Find the value of k.
If we wanted to choose a number m such that P(X \leq m)= 0.9, what region would we expect m to be in?
Hence, calculate m such that P(X \leq m)= 0.9.
The probability density function of a random variable X is defined by \\ p \left( x \right) = k x^{2} for 0 \leq x \leq \sqrt[3]{4}, and p \left( x \right) = 0 elsewhere, is shown:
Find the value of the constant k.
The probability density function of a random variable X is defined by \\ p \left( x \right) = k \sqrt{x} for 0 \leq x \leq 16, and p \left( x \right) = 0 elsewhere, is shown:
Find the exact value of the constant k.
The probability density function of a random variable X is defined by \\ p \left( x \right) = k e^{ - 5 x } for x \geq 0, and p \left( x \right) = 0 elsewhere, is shown:
Find the value of the constant k.
The probability density function of a random variable X is defined by \\ p \left( x \right) = k \sin 2 x for 0 \leq x \leq \dfrac{\pi}{6}, and p \left( x \right) = 0 elsewhere is shown:
Find the value of the constant k.
Calculate P\left( X \lt \dfrac{\pi}{8} \right).
Calculate P\left( X \leq \dfrac{7 \pi}{48} | X \geq \dfrac{\pi}{24} \right). Round your answer to two decimal places.
Consider the probability density function p \left( x \right) shown for a random variable X:
Calculate the area between p \left( x \right) and the x-axis, without using calculus.
Which features of p \left( x \right) are also features of all continuous probability distribution functions?
Calculate the cumulative distribution function F \left( x \right) = \int_{-\infty}^{x} p \left( t \right) dt when \\3.2 \leq x \leq 6.8.
Hence, calculate the following:
Consider the probability density function p \left( x \right) = \dfrac{5}{17} for 1.2 \leq x \leq 4.6 and p \left( x \right) = 0 otherwise.
Sketch the graph of the function.
Calculate the cumulative distribution function F \left( x \right) = \int_{-\infty}^{x} p \left( t \right) dt when 1.2 \leq x \leq 4.6.
Hence, calculate the following:
Consider the probability density function:
p \left( x \right) = \begin{cases} \dfrac{x}{96} & \quad 0 \leq x \leq 12\\ -\dfrac{x}{32} + \dfrac{1}{2} & \quad 12 \lt x \leq 16\\ 0 & \quad \text{otherwise} \end{cases}Sketch the graph of the function.
Use geometric reasoning to calculate the area under p \left( x \right).
State the cumulative distribution function.
A continuous random variable X has probability density function f \left( x \right) = \dfrac{6}{x^{2}} for \left[3, 6\right] and f \left( x \right) = 0 otherwise.
Confirm that \int_{3}^{6} \dfrac{6}{x^{2}} dx = 1.
Calculate the following probabilities:
A continuous random variable X has cumulative distribution function given below:
F\left( x \right) = \begin{cases} 0 & \quad \text{ for } x \lt \dfrac{1}{3}\\ \dfrac{1}{4 \ln 3} \ln(9x^2) & \quad \text{ for } \dfrac{1}{3} \leq x \leq 3 \\ 1 & \quad \text{ for } x\gt3 \end{cases}Calculate P \left( X \gt 2 \right), rounded to two decimal places.
Calculate t such that P \left( X \lt t \right) = \dfrac{1}{4}.
When a pair of jeans is taken in for the hem of the jeans to be shortened, the off-cuts from the leg of the jeans are found to be a random length between 2.6 and 7.5 centimetres. Let X be the length of the off-cut.
State the probability density function for X, p \left( x \right).
State the cumulative probability distribution function for X, F\left( x \right).
The shop assistant randomly chooses an off-cut from a container with off-cuts of at least length 3.7 \text{ cm}. Find the probability that the off-cut they select is no longer than 6 \text{ cm}.
The time taken for viewers to begin watching the latest TV episode of "If You Are The Zero" after it is released was found to be a random time between 3 and 150 minutes. Let X be the time taken for a user to begin watching the new episode of the show.
State the probability density function for X, p \left( x \right).
State the cumulative probability distribution function for X, F\left( x \right).
Calculate the probability that a user who waited at most 30 minutes to start watching the show, waited no longer than 15 minutes.
The length of time X after a computer is turned on until it crashes can be modelled by the probability density function p \left( x \right) = \dfrac{1}{36} e^{ - \frac{x}{36} } when x \geq 0, and p \left( x \right) = 0 otherwise.
State the cumulative probability distribution function for X, F\left( x \right).
Find the probability that the computer crashes after exactly 10 hours of use.
Find the probability that the computer can last at least 50 hours of use. Round your answer to two decimal places.
Find the probability that a computer which has lasted 12 hours will last 30 hours without crashing. Round your answer to two decimal places.
The distance (X) between faults in newly laid cabling can be modelled by the probability density function p \left( x \right) = \dfrac{1}{100} e^{ - \frac{x}{100} } when x \geq 0, and p \left( x \right) = 0 otherwise.
State the cumulative probability distribution function for X, F\left( x \right).
Find the exact probability that the cable has a fault within the first 30 metres of the last fault.
Find the exact probability that the cable does not have a fault within the first 200 metres after the last fault.
Find the probability that if it is known that the fault occurs within the first 190 metres, the fault is found after the first 80 metres. Round your answer to two decimal places.
The mass of a 6 week old puppy, in grams, is modelled by a continuous random variable X which has probability density function p \left( x \right) defined by \\ p \left( x \right)=\begin{cases} k \sin \left( \dfrac{\pi}{240} \left(x - 450\right)\right) & \text{for } 450 \leq x \leq 690 \\ 0 & \text{otherwise} \end{cases}
Find the exact value of k.
Calculate the exact probability that a randomly chosen puppy weighs less than 550 grams.
Calculate the exact probability that a randomly chosen 6 week puppy weighs more than 570 grams, if we are told that it weighs less than 630 grams.
The lifetime of a household washing machine can be modelled by the following probability density function, where x is the time in years. Assume that p \left( x \right) = 0 for x \lt 0.
p \left( x \right) = \begin{cases} kx & \quad \text{ if } 0 \leq x \leq 2\\ kx^{-6} & \quad \text{ if } x \geq 2 \end{cases}Find the value of k.
Find the probability that the washing machine has a lifetime between 0 and 2 years.
In which interval does the median lifetime of a washing machine lie?
Calculate the median lifetime M years of a household washing machine, rounded to two decimal places.
Hence, calculate the probability that a washing machine which has already lasted 2 years, does not need replacing before 4 years have passed.