written 2.6 years ago by
yashbeer
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i) Probability that a randomly selected furnace repair require more than 2 hours
Let x be random variable that denotes the amount of time a repairman needs to fix a furnace. When X is uniformly distributed between (1.5, 4 ) (a, b)
So, required probablity is  $1  P(0 \lt X \lt 2)$
$\begin{aligned}
\text{We know, } P(0 \lt X \lt 2) &= F(2)  F(0) \\
&= \frac{20}{ba} \\
&= \frac{2}{41.5} \\
&= \frac{2}{2.5} \\
&= 0.8
\end{aligned}$
$\therefore$ Probability that a randomly selected furnace repair requires more than two hours = $1  0.8 = 0.2$
ii) Find probability that randomly selected furnace repair requires less than 3 hours
$\begin{aligned}
&=P(\alpha x \lt 3) \\
&=F(3)F(0) \\
&=\frac{30}{ba} \\
&=\frac{3}{41 \cdot 5} \\
&=\frac{3}{2 \cdot 5} \\
&=1 \cdot 2
\end{aligned}$
iii) Find the mean and standard deviation
mean, $E(x)=\frac{a+b}{2}=\frac{1 \cdot 5+4}{2}=\frac{45}{2} = 2.25$
$\begin{aligned} \text { variance, } V(x) &=\frac{(ba)^{2}}{12} \\ &=\frac{(41.5)^{2}}{12}=\frac{(2.5)^{2}}{12} \\ &=\frac{6.25}{12}=0.520 \end{aligned}$
$\begin{aligned} \therefore \text { Standard Deviation } &=\sqrt{\text{Variance}} \\ &=\sqrt{0.520} \\ &=0.721 \end{aligned}$