Problem 15E

Measurements are made on the length and width (in cm) of a rectangular component. Because of measurement error, the measurements are random variables. Let X denote the length measurement and let Y denote the width measurement. Assume that the probability density function of X is

and that the probability density function of Y is

Assume that the measurements X and Y are independent.

a. Find P(X<9.98).

b. Find P(Y > 5.01).

c. Find P(X<9.98 and Y > 5.01).

d. Find μX.

e. Find μγ.

Solution:

Step 1:

Let X denote the length measurement and let Y denote the width measurement. Here the probability density function of X is

And the density function of Y is

We have to find

- P(X<9.98)
- P(Y> 5.01)
- P(X<9.98 and Y>5.01)
- E(X)
- E(Y)

Step 2:

- We have to find the probability P(X<9.98)

P(X<9.98) = f(x) dx

=

= 10 [X

= 10 (9.98-9.95)

= 0.3.

Therefore the probability P(X<9.98) is 0.3.

(b) We have to find the probability that P(Y>5.01)

P(Y>5.01) = f(y) dy

= 5 dy

= 5 [y

= 5 [5.1-5.01]

= 0.45

Therefore the probability P(Y>5.01) is 0.45.

(c) We have to find the probability that P(X<9.98 and Y>5.01)

Since the measurements X and Y are independent

P(X<9.98 and Y>5.01) = P(X<9.98) P(Y>5.01)

= (0.3) (0.45)

= 0.135

Step 3 of 3:

(d) we have to find the mean of X

E(x) = x f(x) dx

= 10 x dx

= 10 [

= 5 ( 2)

= 10

Therefore the mean value of x , E(x) =10

(e) We have to find the mean of Y

E(y) = y f(y)dy

= dy

= 5 [

= 2.5 [2]

= 5

Therefore the mean of Y, E(y)= 5.