Problem 16E

The thickness Xof a wooden shim (in mm) has probability density function

a. Find μX

b. Find

c. Let Y denote the thickness of a shim in inches (1 mm = 0.0394 inches). Find μγ and

d. If three shims are selected independently and stacked one atop another, find the mean and variance of the total thickness.

Solution 16E

Step1 of 5:

We have a random variable X it presents the thickness of wooden shim and it has a probability density function:

Here Our goal is:

a).We need to find

b).We need to find

c).We need to Find and by taking Y as the thickness of a shim in inches (1 mm = 0.0394 inches).

d). If three shims are selected independently and stacked one atop another, find the mean and variance of the total thickness.

Step2 of 5:

a).

=

=

= [therefore ]

=

=

=

=

= -27 - (-32)

= -27 + 32

= 5

Hence, = 5.

Step3 of 5:

b).

=

=

= [therefore ]

=

=

=

=

=

=

= 25.2

Hence, = 25.2.

Now,

= 25.2 -

= 25.2 - 25

= 0.2

Hence, = 0.2.

Step4 of 5:

c).

We have c = 0.0394

= c

= 0.0394(5)

= 0.197

Hence, = 0.197.

=

= 3.1047

Hence, = 3.1047.

Step5 of 5:

d).

Let us consider are three thicknesses of wooden shim.

Let Z = be the total thickness.

Now,

= 3

= 3(5)

= 15

Hence, = 15.

= 3()

= 3(0.2)

= 0.6

Hence, = 0.6.