Random Variable x

[CloudNine]

Hello everyone,

I'm a little bit lost in Probability & Statistic. I need some explanation of the exercises. There is some questions I don't understand.

" There are two machines in the factory which work independently. Probability of failure of the first machine is 0.2 and probability of failure of the other machine is 0.3. The random variable X is defined as the number of machines that are simultaneously out of order. "

What is the probability mass function of X?

Answer: P(X = 0) = 0,8*0,7 = 0,56 / P(X = 1) = 1 - (0,56*0,06) = 0,38 / P(X = 2) = 0,2 * 0,3 = 0,06
At this question, I don't understand how to find the results.

Thank you for your help,

CloudNine,

 

[Romsek]

\(\displaystyle P[0] = P[\text{both machines are working}] = (1-0.2)(1-0.3) = (0.8)(0.7) = 0.56\)

\(\displaystyle P[2] = P[\text{both machines have failed}] = (0.2)(0.3) = 0.06\)

\(\displaystyle
P[1] = P[\text{machine 1 is working and machine 2 has failed}] + P[\text{machine 1 has failed and machine 2 is working}] = \\
(1-0.2)(0.3) + (0.2)(1-0.3) = (0.24)+(0.14) = 0.38\)

Another way of looking at \(\displaystyle P[1]\) is

\(\displaystyle P[1] = 1 - P[2]-P[0] = 1 - 0.56 - 0.06 = 0.38\)

 

[Steven G]

Hello everyone,

I'm a little bit lost in Probability & Statistic. I need some explanation of the exercises. There is some questions I don't understand.

" There are two machines in the factory which work independently. Probability of failure of the first machine is 0.2 and probability of failure of the other machine is 0.3. The random variable X is defined as the number of machines that are simultaneously out of order. "

What is the probability mass function of X?

Answer: P(X = 0) = 0,8*0,7 = 0,56 / P(X = 1) = 1 - (0,56*0,06) = 0,38 / P(X = 2) = 0,2 * 0,3 = 0,06
At this question, I don't understand how to find the results.

Thank you for your help,

CloudNine,

You found your results!
The pmf(X) is of list of probabilities,, one for each value X can take on.
If your example, X can be 0, 1 or 2. So the pmf(x) would be the values for P(X=0), P(X=1) and P(X=2). This is exactly what you did!
However, that does not mean that you got all the correct probabilities!
Why is P(X=1) = 1 - P(X=0)*P(X=1)??
There are exactly 3 outcomes and they do not overlap. So P(X=0) + P(X=1) + P(X=2) = 1 and then P(X=1) = 1- [P(X=0) + P(X=2)]

 

[CloudNine]

Hello everyone !

Thank you Romsek and Jomo for your help ! I finally understand I aced the maths test ! Thank you again !

CloudNine,