The main bearing clearance (in mm) in a certain type of engine is a random variable with probability density function

a. What is the probability that the clearance is less than 0.02 mm?

b. Find the mean clearance.

c. Find the standard deviation of the clearances.

d. Find the cumulative distribution function of the clearance.

e. Find the median clearance.

f. The specification for the clearance is 0.015 to 0.063 mm. What is the probability that the specification is met?

Step 1 of 6 :

The main bearing clearance in a in a certain type of engine is a random variable with the probability density function is

Our goal is to find :

a). What is the probability that the clearance is less than 0.02 mm?

b). Find the mean clearance.

c). Find the standard deviation of the clearance.

d). Find the cumulative distribution function of the clearance.

e). Find the median clearance.

f). Find the probability that the specification is met.

a).

Now we have to find the probability that the clearance is less than 0.02 mm.

So P(X < 0.02) is

The probability density function is

Here we take the limits 0 to 0.02.

Then,

P(X < 0.02) =

We are integrating.

P(X < 0.02) =

P(X < 0.02) =

P(X < 0.02) =

P(X < 0.02) =

P(X < 0.02) =

Therefore the probability that the clearance is less than 0.02 mm is 0.125.

Step 2 of 6 :

b).

Now we have to find the mean clearance.

Mean is

Here the limits is 0 to 0.04 and 0.04 to 0.08.

Or

We integrate the above equation and we substituted that equation.

Then we get the mean value.

Therefore the mean value is 0.04.

Step 3 of 6 :

c).

Now we have to find the standard deviation of the clearance.

We are applying the limits 0 to 0.04 and 0.04 to 0.08.

The variance is

We integrate the above equation and we substituted that equation.

Then we get the variance value.

Standard deviation is the square root of the variance.

Therefore the standard deviation of the clearance is 0.01634.