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8.04 Applications of general continuous random variables

Interactive practice questions

For the average individual, completing their personal income tax return can take between $1$1 and $6$6 hours.

a

The time taken to complete the tax return, where $t$t is the time in hours, can be modelled by the probability density function:

$f\left(t\right)$f(t) $=$= $k\left(t-1\right)\left(6-t\right)\left(t+2\right)$k(t1)(6t)(t+2) if $1\le t\le6$1t6
$0$0 for all other values of $t$t

Calculate the value of $k$k, using the capabilities of your CAS calculator.

b

Calculate the probability, $p$p, that it takes someone exactly $2$2 hours to complete their tax return.

c

Using the capabilities of your CAS calculator, calculate the probability, $q$q, that it takes someone between $2$2 and $3.5$3.5 hours to complete their tax return. Give your answer correct to two decimal places.

d

Let $T$T be the continuous random variable representing the time it takes for an individual to complete their personal income tax return. Use calculus to calculate the expected value of $T$T to two decimal places.

Medium
9min

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.

Medium
4min

The amount of coffee used by a café each week is modelled by a continuous random variable $X$X with a mean of $14.5$14.5 kg and a standard deviation of $1.5$1.5 kg. If the coffee costs $C=24X+15$C=24X+15 in dollars (due to cost per kg and weekly delivery fee), find:

Easy
2min

A written French examination is worth a total of $180$180 marks. The results of the examination can be modelled by a continuous random variable $X$X where the expected value $E\left(X\right)$E(X) is $117$117 and the variance $V\left(X\right)$V(X) is $15$15.

Medium
4min
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Outcomes

4.2.3

identify the expected value, variance and standard deviation of a continuous random variable and evaluate them using technology

4.2.4

examine the effects of linear changes of scale and origin on the mean and the standard deviation

3.3.14

identify contexts suitable for modelling by binomial random variables

3.3.15

determine and use the probabilities P(X=x)=columnvector(nx)p^x(1−p)^(n−x) associated with the binomial distribution with parameters n and p; note the mean np and variance np(1−p) of a binomial distribution

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