The thickness of a washer (in mm) is a random variable with probability density function

a. What is the probability that the thickness is less than 2.5 m?

b. What is the probability that the thickness is between 2.5 and 3.5 m?

c. Find the mean thickness.

d. Find the standard deviation a of the thicknesses.

e. Find the probability that the thickness is within ±σ of the mean.

f. Find the cumulative distribution function of the thickness.

Step 1 of 6 :

The thickness of a washer is a random variable with the probability density function is

Our goal is to find :

a). What is the probability that the thickness is less than 2.5 m.

b). What is the probability that the thickness is between 2.5 and 3.5 m.

c). Find the mean thickness.

d). Find the standard deviation of the thickness.

e). Find the probability that the thickness is within of the mean.

f). Find the cumulative distribution function of the thickness

a).

Now we have to find the probability that the thickness is less than 2.5 m.

The probability density function is

Here we take the limits 2 to 2.5.

P(x < 2.5)

Here we computing the indefinite integral.

Then,

P(x < 2.5)

We calculated the equation.

Then we get

P(x < 2.5)0.781252

P(x < 2.5)

P(x < 2.5)

Therefore the probability that the thickness is less than 2.5 m is 0.2428.

Step 2 of 6 :

b).

Now we have to find the probability that the thickness is between 2.5 and 3.5 m.

The probability density function is

Here we take the limits 2.5 to 3.5.

P(2.5< x < 3.5)

Here we computing the indefinite integral.

Then,

P(2.5< x < 3.5)

We calculated the equation.

Then we get.

P(2.5< x < 3.5)1.2567-0.781252

P(2.5< x < 3.5)1.29567-0.781252

P(2.5< x < 3.5)0.514418

Therefore the probability that the thickness is between 2.5 and 3.5 m is 0.514418.

Step 3 of 6 :

c).

Now we have to find the mean thickness.

Here we take the limits 2 to 4.

Then the mean is

Here we computing the indefinite integral.

Then,

We calculated the equation.

Then we get.

Therefore the mean is 3.