# 概率、统计和数据|MTH1004 Probability, Statistics and Data代写

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for all speeds $v$ between 60 and 90 . We can now obtain the probability density $f_{V}$ of $V$ by differentiating:
$$f_{V}(v)=\frac{\mathrm{d}}{\mathrm{d} v} F_{V}(v)=\frac{\mathrm{d}}{\mathrm{d} v}\left(3-\frac{180}{v}\right)=\frac{180}{v^{2}}$$
for $60 \leq v \leq 90$.
It is amusing to note that with the second model the traffic police write fewer speeding tickets because
$$\mathrm{P}(V>80)=1-\mathrm{P}(V \leq 80)=1-\left(3-\frac{180}{80}\right)=\frac{1}{4}$$

## MTH1004 COURSE NOTES ：

As an example, let $X$ be a random variable with an $N\left(\mu, \sigma^{2}\right)$ distribution, and let $Y=r X+s$. Then this rule gives us
$$f_{Y}(y)=\frac{1}{r} f_{X}\left(\frac{y-s}{r}\right)=\frac{1}{r \sigma \sqrt{2 \pi}} \mathrm{e}^{-\frac{1}{2}((y-r \mu-s) / r \sigma)^{2}}$$
for $-\infty<y<\infty$. On the right-hand side we recognize the probability density of a normal distribution with parameters $r \mu+s$ and $r^{2} \sigma^{2}$. This illustrates the following rule.