A ) Current measurements in a wire follow a normal distribution with a mean of 1

Question

A) Current measurements in a wire follow a normal distribution with a mean of 10 mA and a variance of 4 mA2. What is the probability that a measurement exceeds 13 mA? What is the threshold current, if the threshold current specification for the wire is the current that can only be exceeded 0.5% of the time?

B) Molecules fill a container and move arbitrarily about within its volume. Each molecule’s position is independent of the other molecules. The container can be thought to be divided into right and left halves (an imaginary division without physical walls). On average, half of the molecules will be in each half of the container. However, at any instant in time there is likely to be slightly more molecules in one side or the other. Find the probability that the right side of the container contains 0.49 to 0.51 of the molecules at a given instant in the cases where there are a) 100, b) 1000, c) 10,000, and d) 1,000,000 molecules in the box. Interactions between molecules can be neglected.

Explanation / Answer

a)

Current measurements in a wire follow a normal distribution with a mean of 10 mA and a variance of 4 mA2. What is the probability that a measurement exceeds 13 mA?

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    13      
u = mean =    10      
          
s = standard deviation =    2      
          
Thus,          
          
z = (x - u) / s =    1.5      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.5   ) =    0.066807201 [ANSWER]

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What is the threshold current, if the threshold current specification for the wire is the current that can only be exceeded 0.5% of the time?

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.995      
          
Then, using table or technology,          
          
z =    2.575829304      
          
As x = u + z * s,          
          
where          
          
u = mean =    10      
z = the critical z score =    2.575829304      
s = standard deviation =    2      
          
Then          
          
x = critical value =    15.15165861 mA [ANSWER]  

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