Question

A defective machine in an electronics factory produces rectangular chips whose sides are modeled as independent continuous random variables X and Y. While X has a uniform distribution between 1 and 2, Y has a uniform distribution between 2 and 3. What is the expected area of a chip? 3.75 3.5 3.25 4

Answer 1

Given That

X and Y are sides of the rectangular chip.

This question can be solved using Continuous uniform distribution 

Concept-

Continuous Uniform Distribution often called Rectangular distribution is a family of symmetric probability distribution. It describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters a and b, which are the minimum and maximum values.

The interval can be either be closed or open.

Therefore, the distribution is often abbreviated U(a,b) where U stands for uniform distribution and a & b are parameters.

According to question,

X~U(1,2)

Y~(2,3)

Mean of uniform distribution when X~U(a,b) = a+b/2

So, X = (1+2)/2 = 3/2 = 1.5

Y = (2+3)/2 = 5/2 = 2.5

So, area of triangle = X*Y = 1.5 *2.5 = 3.75

Hence the correct option is a. 3.75

Answer 2

Solution

Given - X and Y are sides of the rectangular chip.

This problem can be solved using Uniform distribution (continuous) defined below-

Continuous Uniform distribution

The continuous Uniform distribution often called Rectangular distribution is a family of symmetric probability distributions. It describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can be either be closed (eg. [a, b]) or open (eg. (a, b)).

Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution and a & b are parameters.

Accprding to question,

X~U(1,2)

Y~U(2,3)

Mean of Uniform dist when X~U(a,b) = (a+b)/2

So, X = (1+2)/2 = 1.5

Similarily, Y = (2+3)/2 = 2.5

Therefore,

Area of Rectangle = X * Y = 1.5 * 2.5 = 3.75

Hence Option (a) is correct.