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For a discrete random variable, the expected value is calculated by summing the product of the value of the random variable and its associated probability, taken over all of the values of the random variable $$ \large \begin{align} E(aX) &= aE(X) \\ E(X+b) &= E(X) + b \\ E(X^2) &\ne \left[E(X)\right]^2 \end{align} $$ Question 1 \( […]
$$ \large \require{AMSsymbols} \displaystyle \begin{align} ^n P_r &= \frac{n!}{(n-r)!} \\ &= \overbrace{n \times (n-1) \times (n-2) \times \cdots }^{r} \end{align} $$ Example \( ^{10} P _3 = \overbrace{10 \times 9 \times 8}^{3} = 720 \) \( \displaystyle ^{10} P _3 = \frac{10!}{(10-3)!} = \frac{10 \times 9 \times 8 \times 7!}{7!} = 10 \times 9 \times […]
Governments raise taxes for services and structural elements such as roads, hospitals and defence. One of the key revenue sources for the government is the income tax. Many businesses and employees pay income tax through their local taxation system. Each pay period, her employer deducts her pay and forwards this to the Taxation Office. Income […]
Bernoulli Trial is an experiment in which the outcome is either a SUCCESS or a FAILURE. SUCCESS means that you are getting the result that you’re counting, but it does not necessarily mean the traditional meaning of triumph or prosperity. The probability of each outcome is independent of the results of the previous trial. The […]
Question 1 Find the Cartesian equation of \( x=t-2, \ y=t^2 \). \( \begin{align} t &= x+2 \\ \therefore y &= (x+2)^2 \end{align} \) Question 2 Find the Cartesian equation of \( x=6y, \ y=3t^2 \). \( \begin{align} \displaystyle t &= \frac{x}{6} \\ y &= 3 \times \left(\frac{x}{6}\right)^2 = 3 \times \frac{x^2}{36} = \frac{x^2}{12} \\ […]
$$ \large \begin{align} a(b+c) &= ab+bc \\ a(b-c) &= ab-bc \end{align} $$ Question 1 Expand \( 3(a+4) \). \( 3(a+4) = 3a+12 \) Question 2 Expand \( 3(a-7) \). \( 3(a-7) = 3a-21 \) Question 3 Expand \( -2(a+5) \). \( -2(a+5) = -2a-10 \) Question 4 Expand \( a(x-2) \). \( a(x-2) = ax-2a […]
Question 1 (a) Find the initial displacement. \( x=0 \) (b) Find the displacement at \( t=4\). \( x=3 \) (c) Find the velocity during \( 0 \le t \le 3 \). \( \displaystyle \text{velocity} = \frac{\text{rise}}{\text{run}} = \frac{3-0}{3-0} = 1 \text{ ms}^{-1} \) (d) Find the velocity during \( 3 […]
$$ \large \displaystyle S_{\infty} = \frac{\text{first term}}{1-\text{common ratio}} $$ Question 1 Find the first three terms of the geometric series for which the common ratio \( r=0.6 \) and \( S_{\infty} = 25 \). \( \displaystyle \begin{align} S_{\infty} &= \frac{a}{1-r} \\ \frac{a}{1-0.6} &= 25 \\ \frac{a}{0.4} &= 25 \\ a &= 25 \times 0.4 \\ […]
Question 1 Find any values of \( x \) for \( y = x^2+x \) if \( y=2 \). \( \begin{align} x^2+x &= 2 \\ x^2+x-2 &= 0 \\ (x-1)(x+2) &= 0 \\ \therefore x &=1 \text{ or } x=-2 \end{align} \) Question 2 Find any values of \( x \) for \( y = […]
Question 1 Simplify the ratio \( 2 \) hours to \( 30 \) minutes. \( \begin{align} 2 \text{ hours} : 30 \text{ minutes} &= 2 \times 60 \text{ minutes} : 30 \text{ minutes} \\ &= 120 : 30 \\ &= 4 : 1 \end{align} \) Question 2 Simplify the ratio \( 50 \) cm to […]