91Ó°ÊÓ

Variables are the terms that may have different values. Here in this distribution variable X is uniformly distributed between the 52 weeks of the year.

a. Uniform of all data

$X~U(1,53)$

b. The height of probability distribution

$$\begin{gathered} f\left(x\right)=\frac{1}{b-a} \\ =\frac{1}{53-1} \\ =\frac{1}{52} \end{gathered}$$

The graph of distribution is

c. The formulation of,

$\begin{array}{rcl}f\left(x\right)& =& \left\{\begin{array}{ll}\frac{1}{b-a}& ,a<X<b\\ 0& ,1<X<53\end{array}\right.\\ f\left(x\right)& =& \left\{\begin{array}{ll}\frac{1}{52}& ,1<X<53\\ 0& ,Otherwise\end{array}\right.\end{array}$

d. The mean can be calculated as;

$\begin{array}{rcl}\mathrm{μ}& =& \frac{a+b}{2}\\ & =& \frac{1+53}{2}\\ & =& 27\end{array}$

Hence the mean is 27

e. The standard deviation

$$\begin{gathered} \mathrm{σ}=\sqrt{\frac{{\left(b-a\right)}^2}{12}} \\ \mathrm{σ}=\sqrt{\frac{{\left(53-1\right)}^2}{12}} \\ \mathrm{σ}=\sqrt{225.33} \\ \mathrm{σ}=15.01 \end{gathered}$$

The standard Deviation is 15.01

f. For continuous probability, the point value is zero.

Hence the value of $P(x=19)=0$

g. The required probability is

$\begin{array}{rcl}P(2& <& x<31)=Base×Height\\ & =& (32-2)×\frac{1}{52}\\ & =& 29×\frac{1}{52}\\ & =& 0.5577\end{array}$

h.

The value of

localid="1649333150490" $\begin{array}{rcl}P(x& >& 40)=Base×Height\\ & =& (53-40)×\frac{1}{52}\\ & =& 13×\frac{1}{52}\\ & =& 0.25\end{array}$

i. The probability of

$\begin{array}{rcl}& & P(12<x\overline{)x<28})=\frac{P(12<x<28)}{P(x<28)}\\ & =& \frac{(28-12)×\frac{1}{52}}{(28-1)×\frac{1}{52}}\\ & =& \frac{16}{27}\\ & =& 0.5926\end{array}$

j. For the 70 percentile

$\begin{array}{rcl}P(x& <& K)= Base×Height\\ 0.70& =& (K-1)×\frac{1}{52}\\ k-1& =& 0.70×52\\ K& =& 36.4+1\\ & =& 37.4\end{array}$

k. The required value can be calculated by calculating the 75th percentile of the data, which can be calculated as below;

$\begin{array}{rcl}P(x& <& K)= Base×Height\\ 0.75& =& (K-1)×\frac{1}{52}\\ k-1& =& 0.75×52\\ K& =& 39+1\\ & =& 40\end{array}$

The value of the minimum of the upper quarter is 40.