Which of the following diagrams show two distributions with the same expectation?
State whether the following scenarios could be modelled by the two distributions from part (a). Explain your answers.
The length of two species of fish A and B which have the same average length, but where the length of species B varies more than the length of species A.
The battery life of two different brands of phones A and B, where phone A has a longer average battery life but the variation in battery life among the two phones is the same.
Which of the following diagrams show two distributions with the same variance?
State whether the following scenarios could be modelled by the two distributions from part (a). Explain your answers.
The wing span of two species of eagles A and B, where species B has a longer average wing span than species A, but there is more variation in wing span amongst species A.
The heights of two tree species A and B, where species B is on average taller than species A, but the variation in height amongst the two species is the same.
The results of two tests A and B where the mean result of both tests was the same, and the spread of the results in both tests was the same.
For each probability density function, find the expected value of a random variable X if it is distributed according to p \left( x \right):
p \left( x \right) = \dfrac{1}{40} where 10 \leq x \leq 50
p \left( x \right) = \dfrac{3}{160} \left(x^{2} + 4 x\right) where 0 \leq x \leq 4
For each of the following probability density functions graphed:
Define the function p \left( x \right).
Find the expected value of a random variable X if it is distributed according to p \left( x \right).
Consider the following probability density functions:
Find the value of k.
Find the expected value of a random variable X if it is distributed according to p \left( x \right).
\\ p \left( x \right)=\begin{cases}k x^{2} & \text{for } 4 \leq x \leq 7 \\ 0 & \text{otherwise} \end{cases}
p \left( x \right) = k \left(2 - x\right) \left(x - 6\right) when 2 \leq x \leq 6
p \left( x \right) = k x^{3} when 1 \leq x \leq 5
Consider the probability density function:
Define the function p \left( x \right).
Use integration to calculate the expected value of a random variable X if it is distributed according to p \left( x \right).
Use integration to find the median value, m, of p \left( x \right).
Consider the probability density function p, where p \left( x \right) \gt 0 when 4 \leq x \leq 12 and p \left( x \right) = 0 otherwise. The graph of y = p \left( x \right) is shown:
Find the expression that represents p \left( x \right) on the domain 4 \leq x \leq 12.
Find the expected value of p \left( x \right).
By performing an integration similar in part (b), we can find that E \left(X^{2}\right) = \dfrac{368}{5}.
Hence, calculate the variance, V \left(X\right).
The probability density function of a random variable X is drawn below. Its non-zero values lie in the region 0 \leq x \leq k.
Calculate the value of k.
What equation defines the probability distribution function of X in the domain 0 \leq x \leq k?
Calculate the following using technology or otherwise:
Expected value of X.
Variance of X.
Consider the probability density function p \left( x \right) = \dfrac{1}{72} \left(x + 4\right) when - 4 \leq x \leq 8.
Use integration to calculate the expected value of a random variable X if it is distributed according to p \left( x \right).
Use integration to find the median value, m, of p \left( x \right).
Consider the probability density function p \left( x \right) = k e^{ - k x } when 0 \leq x.
Find the value of k given that the median value is 2 \ln 2.
Using the product rule, find \dfrac{d}{d x}\left( - x e^{ - 0.5 x } \right).
Find the expected value of a random variable X if it is distributed according to p \left( x \right).
Consider the probability density function p \left( x \right) = 0.25 e^{ - 0.25 x } when 0 \leq x.
Using the product rule, find \dfrac{d}{d x}\left( - x e^{ - 0.25 x } \right).
Use integration and the result from part (a) to calculate the expected value of a random variable X if it is distributed according to p \left( x \right).
Use integration to find the median value, m, of p \left( x \right).
Consider the probability density function\\ p \left(x \right)=\begin{cases}\dfrac{1}{20} & \text{for } 25 \leq x \leq 45 \\ 0 & \text{otherwise} \end{cases}
Find the expected value of p \left( x \right).
Find the variance of p \left( x \right).
Consider the probability density function p, where p \left( x \right) = \dfrac{1}{18} \left(x + 2\right) when - 2 \leq x \leq 4 and p \left( x \right) = 0 otherwise.
Calculate the expected value of a random variable X if it is distributed according to p(x), using technology or otherwise.
Calculate the standard deviation of X.
Consider the probability density function p, where p \left( x \right) = \dfrac{3}{160} \left(x^{2} + 4 x\right) when 0 \leq x \leq 4 and p \left( x \right) = 0 otherwise.
Calculate the expected value of a random variable X if it is distributed according to p \left( x \right), using technology or otherwise.
Calculate the variance, \text{Var} \left(X\right).
Consider the probability density function p, where p \left( x \right) = k x^{2} when 5 \leq x \leq 8 and p \left( x \right) = 0 otherwise.
Integrate \int_{5}^{8} x^{2} dx.
Find k.
Calculate the expected value of a random variable X if it is distributed according to \\ p \left( x \right), using technology or otherwise.
Calculate the variance, \text{Var} \left(X\right).
Consider the probability density function p, where p \left( x \right) = k \cos \left( \dfrac{\pi}{2} x\right) when 0 \leq x \leq 1 and p \left( x \right) = 0 otherwise.
Integrate \int_{0}^{1} \cos \left( \dfrac{\pi}{2} x\right) dx.
Find k.
Calculate the expected value, E \left(X\right), using technology or otherwise.
Calculate the standard deviation, rounded to two decimal places.
A continuous random variable X has a uniform probability density function over the domain \left[10, 80\right]. X is transformed to the random variable Y by Y = 2 X + 3. State the domain in which Y exists.
A uniform probability density function, P \left( x \right), is positive over the domain \left[20, 50\right] and 0 elsewhere.
Define the function P(x) for this distribution.
Use integration to calculate the expected value of the distribution.
Use integration to calculate the variance V \left( X \right) of the distribution.
The distribution is transformed to the random variable Y by Y = 2 X + 4. Calculate E \left( Y \right), the expected value of Y.
Calculate the variance V \left( Y \right) of Y.
Calculate the standard deviation of Y, rounded to one decimal place.
A continuous random variable X has a uniform probability density function, which is positive over the domain \left[20, 80 \right]. Y is a transformation of X, given by Y = 2 X + 5. Calculate the following:
E \left(X\right)
\text{Var} \left(X\right)
E \left(Y\right)
\text{Var} \left(Y\right)
Standard deviation of Y
Consider the graph of the probability density function P \left( x \right):
Define the function p(x) for this distribution.
Use integration to calculate the expected value of the distribution.
Use integration to calculate the variance V \left( X \right) of the distribution.
The distribution is transformed to the random variable Y by Y = 5 - 2 X. Calculate E \left( Y \right), the expected value of Y.
Calculate the variance of Y.
Consider the graph of the probability density function P \left( x \right):
State the function defining this distribution.
Use integration to calculate the expected value of the distribution.
Given that E \left( X^{2} \right) = \dfrac{350}{3}, calculate the variance V \left( X \right) of the random variable X.
The distribution is transformed to the random variable Y by Y = 3 X + 13. Calculate E \left( Y \right), the expected value of Y.
Calculate the variance of Y.
The probability density function of a random variable X and its graph are given below:
p \left( x \right) = \begin{cases} kx^2 & \text{ if } 0 \leq x\leq 8 \\ 0 & \text{for all other values of } x \end{cases}Find the value of k.
Use integration to calculate the expected value of the distribution.
Use integration to calculate the variance V \left( X \right) of the distribution.
The distribution is transformed to the random variable Y by Y = 0.5 X - 1. Calculate E \left( Y \right), the expected value of Y.
Calculate the variance of Y.
The probability density function of a random variable X and its graph are given below:
p \left( x \right) = \begin{cases} k\sin x & \text{if } 0 \leq x\leq \pi \\ 0 & \text{for all other values of }x \end{cases}Find the value of k.
Find the derivative of \sin x - x \cos x.
Hence, calculate the expected value of the distribution. Express your answer in exact form.
The distribution is transformed to the random variable Y by Y = 11 - 2 X. Calculate E \left( Y \right), the expected value of Y in terms of \pi.
Students are asked to use a protractor to measure out an angle of 60 \degree. The teacher finds that the angles drawn are uniformly distributed between 57 \degree and 63 \degree. Let X be the size of an angle drawn and p \left( x \right) the probability density function of X.
Sketch the graph of p \left( x \right) on a coordinate plane.
Define the function p \left( x \right).
Calculate the expected value of a random variable X if it is distributed according to p \left( x \right).
The length of a newborn baby, in centimeters, is a continuous random variable X which is defined by the probability density function p \left( x \right) = k \sin \left( \dfrac{\pi}{25} \left(x - 45\right)\right) when 45 \leq x \leq 70.
Calculate the expected length of a randomly selected newborn.
The length of time, in minutes, between customers calling a help centre can be modelled by the probability density function p \left( x \right) = k e^{ - k x } when 0 \leq x.
Find the expected value of a random variable X if it is distributed according to p \left( x \right).
Mail gets delivered to an office any time between 10:15 am and 10:55 am each day. The probability that the mail arrives at any particular time in this period is the same. Let X be the time after 10:15 am that the mail is delivered:
Define the probability distribution function, p for the random variable X.
Find the probability that the mail is delivered to the office after 10:30 am.
Find the probability the mail is delivered before 10:50 am, given that it was delivered after 10:25 am.
Find the expected value of p \left( x \right).
Find the variance of p \left( x \right).
On a Sunday morning in the emergency department, the time, x hours, between arriving and being seen by a doctor can be modelled by the following probability density function:
p \left( x \right) = \begin{cases} \dfrac{4}{\pi(1+x^2)} & \text{if } 0 \leq x\leq 1 \\ 0 & \text{otherwise} \end{cases}Use integration to calculate the expected wait time for a patient who arrives on a Sunday morning. Round your answer to one decimal place.
Find the expected wait time in seconds to one decimal place.
Given that the variance of the distribution in hours is \left(\dfrac{4}{\pi} - 1\right), calculate the standard deviation, rounded to one decimal place, of the distribution in seconds.
An online support centre receives requests for assistance through instant messaging. The time between requests for support on a Sunday morning, in time t seconds can be modelled by the random variable T with a probability function p, where p \left( t \right) = \dfrac{1}{10} e^{ - \frac{t}{10} } for all t \geq 0 and p \left( t \right) = 0 otherwise.
Find the probability, rounded to two decimal places, that the time between two requests is less than 5 seconds.
If the time between requests is more than 7 seconds, find the probability, rounded to two decimal places, that the time is less than 12 seconds.
Calculate the following:
Expected time between requests.
Standard deviation of T.
Expected value if the time was recorded in minutes.
Standard deviation if the time was recorded in minutes.
Driving from Sydney to Gosford takes somewhere between 60 and 100 minutes. A random variable X is the number of minutes (in excess of 60 minutes) that it takes to make the trip from Sydney to Gosford.
Consider the probability density function p, where the values of p \left( x \right) are those shown below:
p \left( x \right) = \begin{cases} \dfrac{1}{400}x & \text{when } 0 \leq x\leq 20 \\ -\dfrac{1}{400}x+\dfrac{1}{10}; & 20\lt x\leq 40\\ 0 & \text{otherwise} \end{cases}State the expected number of minutes the trip takes.
Calculate the following:
Probability that a trip from Sydney to Gosford will take more than 90 minutes.
Probability that the trip will take between 70 and 80 minutes.
Variance of X.