Answer:
Hence, the list ten different possible outcomes are
, the probability distribution of X are
and the cumulative distribution function is ![F(x)=\left\{\begin{array}{lc}0 & x](https://tex.z-dn.net/?f=F%28x%29%3D%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Blc%7D0%20%26%20x%3C0%20%5C%5C0.3%20%26%200%20%5Cleq%20x%3C1%20%5C%5C0.9%20%26%201%20%5Cleq%20x%3C2%20%5C%5C1%20%26%202%20%5Cleq%20x%5Cend%7Barray%7D%5Cright.)
Step-by-step explanation:
(a)
List the ten different possible outcomes.
The number of ways that two boards are drawn from each lot of
lots are ![{5 \choose 2}](https://tex.z-dn.net/?f=%7B5%20%5Cchoose%202%7D)
![{5 \choose 2}=\frac{5!}{2!(5-2)!}](https://tex.z-dn.net/?f=%7B5%20%5Cchoose%202%7D%3D%5Cfrac%7B5%21%7D%7B2%21%285-2%29%21%7D)
![\Rightarrow {5 \choose 2}=\frac{5\times 4\times 3!}{2!\times 3!}](https://tex.z-dn.net/?f=%5CRightarrow%20%7B5%20%5Cchoose%202%7D%3D%5Cfrac%7B5%5Ctimes%204%5Ctimes%203%21%7D%7B2%21%5Ctimes%203%21%7D)
![\Rightarrow {5 \choose 2}=\frac{5\times 4}{2!}](https://tex.z-dn.net/?f=%5CRightarrow%20%7B5%20%5Cchoose%202%7D%3D%5Cfrac%7B5%5Ctimes%204%7D%7B2%21%7D)
![\Rightarrow {5 \choose 2}=\frac{20}{2}](https://tex.z-dn.net/?f=%5CRightarrow%20%7B5%20%5Cchoose%202%7D%3D%5Cfrac%7B20%7D%7B2%7D)
![\Rightarrow {5 \choose 2}=10](https://tex.z-dn.net/?f=%5CRightarrow%20%7B5%20%5Cchoose%202%7D%3D10)
So, the
combinations are as follows :
![(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)](https://tex.z-dn.net/?f=%281%2C2%29%2C%281%2C3%29%2C%281%2C4%29%2C%281%2C5%29%2C%282%2C3%29%2C%282%2C4%29%2C%282%2C5%29%2C%283%2C4%29%2C%283%2C5%29%2C%284%2C5%29)
(b)
The aim is to find the probability distribution when two boards are chosen at random and board's one and two is the only defective boards.
Let X be the number of defective boards in the lot.
Therefore, select the combinations of the boards that are without 1 and 2 from 10 combinations.
Compute the ![P(X=0)](https://tex.z-dn.net/?f=P%28X%3D0%29)
![P(X=0)=P\{(3,4),(3,5),(4,5)\}](https://tex.z-dn.net/?f=P%28X%3D0%29%3DP%5C%7B%283%2C4%29%2C%283%2C5%29%2C%284%2C5%29%5C%7D)
![=\frac{3}{10}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B3%7D%7B10%7D)
![=0.3](https://tex.z-dn.net/?f=%3D0.3)
The probability that defectives did not occur in the lot is 0.3
Compute the ![P(X=1)](https://tex.z-dn.net/?f=P%28X%3D1%29)
Select the combinations of the boards that are with one defective either 1 and 2 from the 10 combinations.
![P(X=1)=P\{(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)\}](https://tex.z-dn.net/?f=P%28X%3D1%29%3DP%5C%7B%281%2C3%29%2C%281%2C4%29%2C%281%2C5%29%2C%282%2C3%29%2C%282%2C4%29%2C%282%2C5%29%5C%7D)
![=\frac{6}{10}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B6%7D%7B10%7D)
![=0.6](https://tex.z-dn.net/?f=%3D0.6)
The probability that one defectives occurred in the lot is 0.6.
Compute the ![P(X=2)](https://tex.z-dn.net/?f=P%28X%3D2%29)
Select the combinations of the boards that are with two defective 1 and 2 from the 10 combinations.
![P(X=2)=P\{(1,2)\}](https://tex.z-dn.net/?f=P%28X%3D2%29%3DP%5C%7B%281%2C2%29%5C%7D)
![=\frac{1}{10}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B10%7D)
![=0.1](https://tex.z-dn.net/?f=%3D0.1)
The probability that two defectives occurred in the lot is 0.1.
(c)
The cumulative distribution function (cdf)
is defined as,
![\begin{aligned}F(X&=x)=P(X \leq x) \\F(0) &=P(X \leq 0) \\&=P(X=0) \\&=0.3 \\F(1) &=P(X \leq 1) \\&=P(X=0)+P(X=1) \\&=0.3+0.6 \\&=0.9 \\F(2) &=P(X \leq 2) \\&=P(X=0)+P(X=1)+P(X=2) \\&=0.3+0.6+0.1 \\&=1.0\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7DF%28X%26%3Dx%29%3DP%28X%20%5Cleq%20x%29%20%5C%5CF%280%29%20%26%3DP%28X%20%5Cleq%200%29%20%5C%5C%26%3DP%28X%3D0%29%20%5C%5C%26%3D0.3%20%5C%5CF%281%29%20%26%3DP%28X%20%5Cleq%201%29%20%5C%5C%26%3DP%28X%3D0%29%2BP%28X%3D1%29%20%5C%5C%26%3D0.3%2B0.6%20%5C%5C%26%3D0.9%20%5C%5CF%282%29%20%26%3DP%28X%20%5Cleq%202%29%20%5C%5C%26%3DP%28X%3D0%29%2BP%28X%3D1%29%2BP%28X%3D2%29%20%5C%5C%26%3D0.3%2B0.6%2B0.1%20%5C%5C%26%3D1.0%5Cend%7Baligned%7D)
Therefore, the cumulative distribution function (cdf)
is,
![F(x)=\left\{\begin{array}{lc}0 & x](https://tex.z-dn.net/?f=F%28x%29%3D%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Blc%7D0%20%26%20x%3C0%20%5C%5C0.3%20%26%200%20%5Cleq%20x%3C1%20%5C%5C0.9%20%26%201%20%5Cleq%20x%3C2%20%5C%5C1%20%26%202%20%5Cleq%20x%5Cend%7Barray%7D%5Cright.)