Question

The​ random-number generator on calculators randomly generates a number between 0 and 1. The random variable​ X, the number​ generated, follows a uniform probability distribution. ​(a) Identify the graph of the uniform density function. ​(b) What is the probability of generating a number between 0.64 and 0.86​? ​(c) What is the probability of generating a number greater than 0.84​?

Answer 1

Part (a)

The graph will be a rectangle spanning from x = 0 to x = 1 along the horizontal portion. The vertical portion spans from y = 0 to y = 1.

The rectangle is 1 by 1. Technically the better name is a square, but I'll stick to rectangle because all uniform distribution graphs are rectangles (only in special cases will there be a square). Any square is a rectangle but not vice versa.

Note how the area is 1 to represent 100% of all possible cases.

Answer: Rectangle spanning from x = 0 to x = 1; also spanning from y = 0 to y = 1.

=======================================================

Part (b)

The gap from 0.64 to 0.86 is 0.22 (because 0.86 - 0.64 = 0.22)

The gap from 0 to 1 is 1

Divide those gap values to get 0.22/1 = 0.22

The probability of picking a random number between 0.64 and 0.86 is 0.22

Answer: 0.22

=======================================================

Part (c)

The gap from 0.84 to 1 is 0.16 because 1-0.84 = 0.16

Divide this over the gap from 0 to 1, to get 0.16/1 = 0.16

Answer: 0.16