Question

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

Answer 1

The random variables X and Y represent the length and width measurements, respectively, of a rectangular component. They are continuous random variables because they are measured quantities rather than discrete countable values. To find the probability of a certain range of values for X or Y, you need to compute the area under the probability density function (pdf) curve between those values.

Step-by-step explanation:

The random variables X and Y represent the length and width measurements, respectively, of a rectangular component.

They are continuous random variables because they are measured quantities rather than discrete countable values.

To find the probability of a certain range of values for X or Y, you need to compute the area under the probability density function (pdf) curve between those values. This can be done by integrating the pdf over the desired interval.

In this case, since X and Y are continuous random variables, you should use a continuous probability distribution such as the normal distribution or the uniform distribution, depending on the characteristics of the measurements.

Answer 2

a. To find P(X < 9.98), calculate the area under f(x) from negative infinity to 9.98. b. To find P(Y > 5.01), calculate the area under g(y) from 5.01 to infinity. c. To find P(X < 9.98 and Y > 5.01), calculate the joint probability as the product of the individual probabilities. d. The mean u can be calculated using the formula u = E(X) + E(Y).

Step-by-step explanation:

a. To find P(X < 9.98), we need to calculate the area under the probability density function f(x) from negative infinity to 9.98.

The function f(x) is a constant 10 between 9.95 and 10.05, and 0 otherwise. So the area under f(x) from negative infinity to 9.98 is 10 * (9.98 - 9.95) = 0.30.

Therefore, P(X < 9.98) = 0.30.

b. To find P(Y > 5.01), we need to calculate the area under the probability density function g(y) from 5.01 to infinity.

The function g(y) is a constant 5 between 4.9 and 5.1, and 0 otherwise. So the area under g(y) from 5.01 to infinity is 5 * (infinity - 5.01) = infinity.

Therefore, P(Y > 5.01) = infinity.

c. To find P(X < 9.98 and Y > 5.01), we need to calculate the joint probability of X < 9.98 and Y > 5.01.

Since X and Y are independent, the joint probability is simply the product of the individual probabilities: P(X < 9.98) * P(Y > 5.01) = 0.30 * infinity = infinity.

d. The mean, denoted as u, can be calculated using the formula: u = E(X) + E(Y), where E(X) is the mean of X and E(Y) is the mean of Y.

Since X and Y are independent, the mean of X is the expected value of X, which can be found by calculating the area under the probability density function f(x) multiplied by x.

The expected value of X is given by: E(X) = integral(10x, 9.95, 10.05) = 10 * (10.05 - 9.95) = 0.10.

Similarly, the expected value of Y is given by: E(Y) = integral(5y, 4.9, 5.1) = 5 * (5.1 - 4.9) = 0.20.

Therefore, the mean u = E(X) + E(Y) = 0.10 + 0.20 = 0.30.

the complete Question is given below:

Measurements are made on the length and width (in cm) of a rectangle 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 is

f(x)= {10 9.95 < x < 10.05 , 0 otherwise

and that the probability density function of Y is

g(y)= { 5 4.9 < y <5.1 , 0 otherwise

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 u (mean)