Before we apply what we have learned to real world situations, let's review some of the important concepts we need from previous chapters that will be useful to recall here.
A probability density function, $f(x)$f(x), must satisfy the following two properties:
- $f(x)\ge0$f(x)≥0 for all $x$x (since probability values are non-negative)
-
$\int_{-\infty}^{+\infty}\ f(x)\ dx=1$∫+∞−∞ f(x) dx=1 (This is because the sum of all the probabilities is $1$1).
Note: Often our probability function occurs between two specific values, on an interval $[a,b]$[a,b] and can be defined as $0$0 elsewhere, thus from the second property above we would have $\int_b^a\ f(x)\ dx=1$∫ab f(x) dx=1.
For a continuous random variable $X$X then the cumulative distribution function (CDF) is:
$F\left(x\right)=P\left(X\le x\right)$F(x)=P(X≤x) for all x.
This means that:
$F(x)=\int_{-\infty}^x\ f(t)\ dt$F(x)=∫x−∞ f(t) dt where $f\left(t\right)$f(t) is the probability density function.
Note: if $f\left(t\right)>0$f(t)>0 on an interval $[a,b]$[a,b], and $0$0 elsewhere, then $F(x)=\int_{-\infty}^x\ f(t)\ dt=\int_a^x\ f(t)\ dt$F(x)=∫x−∞ f(t) dt=∫xa f(t) dt.
The mean, $\mu$μ or $E\left(X\right)$E(X), of a PDF is calculated by:
$E(X)=\int_a^b\ xf\left(x\right)\ dx$E(X)=∫ba xf(x) dx
The median of a PDF is calculated by determining the value of $x$x over the interval $\left[a,b\right]$[a,b] such that the area under $f\left(x\right)$f(x) is equal to $0.5$0.5:
$\int_a^x\ f(x)\ dx=0.5$∫xa f(x) dx=0.5
The variance of a PDF is calculated by:
$Var\left(X\right)=\sigma^2=\int_{-\infty}^{\infty}\ \left(x-\mu\right)^2\ p(x)\ dx$Var(X)=σ2=∫∞−∞ (x−μ)2 p(x) dx
$=\int_a^b\ \left(x-\mu\right)^2\ f(x)\ dx$=∫ba (x−μ)2 f(x) dx
$=\int_a^b\ x^2\ f(x)\ dx-\mu^2$=∫ba x2 f(x) dx−μ2
Before we look at an example and how to best utilise our CAS for these questions, it's useful to note that now we've explored both discrete random variables and continuous random variables, we often see a reappearance of discrete random variables, and in particular the binomial distribution, appearing as subparts in the problems we come across. This is a good time to go back and review the binomial distribution before venturing further!
Select your brand of calculator below to work through an example of an applied question together with calculator instructions.
Casio ClassPad
How to use the CASIO Classpad to complete the following tasks regarding an application of continuous random variables.
The time it takes in minutes for a student to complete a puzzle is a random variable $X$X with a probability density function given by:
| $f\left(x\right)$f(x) | $=$= |  |
$\frac{20x-x^2}{1125}$20x−x21125 | if $5\le x\le20$5≤x≤20 | ||
| $0$0 | for all other values of $x$x |
Determine $F\left(x\right)$F(x), the cumulative distribution function of $f\left(x\right)$f(x).
Determine the probability that a student took less than $15$15 minutes to complete the puzzle given that they took at least $8$8 minutes to complete the puzzle.
Give the answer correct to $4$4 decimal places.
Calculate the probability that, of the $40$40 students completing the puzzle, more than $15$15 of them take less than $10$10 minutes to complete the puzzle.
Give the answer correct to $4$4 decimal places.
TI Nspire
How to use the TI Nspire to complete the following tasks regarding an application of continuous random variables.
The time it takes in minutes for a student to complete a puzzle is a random variable $X$X with a probability density function given by:
| $f\left(x\right)$f(x) | $=$= | |
$\frac{20x-x^2}{1125}$20x−x21125 | if $5\le x\le20$5≤x≤20 | ||
| $0$0 | for all other values of $x$x |
Determine $F\left(x\right)$F(x), the cumulative distribution function of $f\left(x\right)$f(x).
Determine the probability that a student took less than $15$15 minutes to complete the puzzle given that they took at least $8$8 minutes to complete the puzzle.
Give the answer correct to $4$4 decimal places.
Calculate the probability that, of the $40$40 students completing the puzzle, more than $15$15 of them take less than $10$10 minutes to complete the puzzle.
Give your answer correct to four decimal places.
Practice questions
question 1
The time $t$t, in hours, that an owl spends hunting each night can be modelled by a continuous random variable with probability density function given below.
| $f\left(t\right)$f(t) | $=$= |  |
$\frac{k}{32}t\left(4-t\right)$k32t(4−t), $0\le t\le4$0≤t≤4 |
| $0$0 otherwise |
Determine the value of $k$k.
Calculate the probability the owl spends more than $3$3 hours hunting during one night.
Calculate the expected time the owl spends hunting in hours.
Calculate the standard deviation of the time the owl spends hunting in hours.
Round your answer to two decimal places.
question 2
In a study on how distracting notifications on a mobile phone can be, individuals were monitored for their reaction time to a notification on their phone while they were at work.
The probability that an individual takes $t$t seconds to respond is modelled by the following probability density function :
$f\left(t\right)$f(t) $=$= 
$k\left(t-1\right)^2e^{-\left(t-1\right)}$k(t−1)2e−(t−1) if $1\le t\le33$1≤t≤33 $0$0 for all other values of $t$t Calculate the value of $k$k to two decimal places, using the capabilities of your CAS calculator.
Using your rounded solution from part (a) and the capabilities of your CAS calculator, calculate the probability $p$p that an individual reacts within the first $4$4 seconds. Round your answer to two decimal places.
Calculate the probability, $q$q, that an individual reacts within $8$8 seconds given that they took at least $7$7 seconds to respond. Round your answer to two decimal places.
Question 3
Due to an outbreak of disease and a shortage of supply, growers predict the prices of bananas, $X$X, to be anywhere between $\$5.90$$5.90 and $\$9.30$$9.30 per kilogram in the next $6$6 months, with all prices equally likely. We are to model $X$X as a continuous random variable.
Let $p\left(x\right)$p(x) be the probability density function for the random variable $X$X.
State the function defining this distribution.
$p\left(x\right)$p(x) $=$= 
$\editable{}$ if $5.90\le x\le9.30$5.90≤x≤9.30 $\editable{}$ for all other values of $x$x Calculate the expected price of bananas in $6$6 months time.
If the predicted price is known to be at least $\$6.90$$6.90 per kilogram, calculate the probability that it’s at most $\$7.60$$7.60 per kilogram.
In $6$6 months time, the price of bananas at $10$10 grocers around the country will be studied. What is the probability that exactly $6$6 grocers are charging more than $\$7.70$$7.70 per kilogram? Round your answer to three decimal places.
What is the probability that at least $2$2 grocers of the $10$10 are charging more than $\$7.70$$7.70? Round your answer to three decimal places.
question 4
The time $t$t, in hours, that a cow grazes in a paddock per day can be modelled by a continuous random variable with a cumulative distribution function given below.
| $F\left(t\right)$F(t) | $=$= | |
$0$0, $t<0$t<0 |
|
$\frac{1}{288}\left(6t^2-\frac{t^3}{3}\right)$1288(6t2−t33), $0\le t\le12$0≤t≤12 |
|||
| $1$1, $t>12$t>12 |
Calculate the probability that the cow grazes for less than $4$4 hours in a given day.
Calculate the probability that the cow grazes for at least $7$7 hours in a given day.
Calculate the median time $m$m, in hours, that the cow spends grazing.