Question

Suppose that the value of a stock varies each day from $\backslash (16\mathrm{to}\backslash )25$with a uniform distribution.

a. Find the probability that the value of the stock is more than $\backslash (19$.

b. Find the probability that the value of the stock is between role="math" localid="1648188993020" $\backslash )19\mathrm{and}\backslash (22$.

c. Find the upper quartile - $25\%$of all days the stock is above what value? Draw the graph.

d. Given that the stock is greater than $\backslash )18$, find the probability that the stock is more than $\$21$.

Short answer

a. The probability that the value of the stock is more than $\$19$is $0.67$

b. The probability that the value of the stock is between $\$19\mathrm{and}\$22$is $0.33$.

c. The upper quartile $-25\%$of all days the stock is above $22.75$.

d. The probability that the stock is more than $\$21\mathrm{is}0.5714$.

Step-by-step solution

Defining the height of Probability distribution

It is given that$X~U(16,25)$

Therefore, the height or probability distribution of the data will be:

$$\begin{gathered} f\left(x\right)=\frac{1}{b-a} \\ =\frac{1}{25-16} \\ =\frac{1}{9} \end{gathered}$$

Finding the probability that the value of the stock is more than $19.

a) The required probability can be calculated as below:

$$\begin{gathered} P(x>19)=base\times height \\ =(25-19)\times \frac{1}{9} \\ =\frac{6}{9} \\ =0.67 \end{gathered}$$

Finding the probability that the value of the stock is between $19 and $22.

b) The required probability can be calculated as below:

$$\begin{gathered} P(19<x<22)=base\times height \\ =(22-19)\times \frac{1}{9} \\ =\frac{3}{9} \\ =0.33 \end{gathered}$$

Finding the upper quartile - 25% of all days the stock is above what value

c) The upper quartile can be calculated as below :

$$\begin{gathered} P(x>k)=base\times height \\ \Rightarrow 0.25=(25-k)\times \frac{1}{9} \\ \Rightarrow 9\times 0.25=25-k \\ \Rightarrow k=22.75 \end{gathered}$$

Finding the probability that the stock is more than $21,given that the stock is greater than $18,

d) The required probability can be calculated as below:

$$\begin{gathered} P(x>21\overline{)x>18})=base\times height \\ =(25-21)\times \frac{1}{(25-18)} \\ =4\times \frac{1}{7} \\ =0.5714 \end{gathered}$$