Question

The random variable z is known to be uniformly distributed between 1 and 1.5. a. Which of the following graphs accurately represents this probability density function? 1. [F(x) 0.25 05 0.75 1.25 1.5 1.75 x 2. (f(x) 0.25 05 0.75 1 1.25 1.5 1.75 2 2 x 3. [f(x) N 0.25 0.5 0.75 1.25. 15. 1.75... 2x - Select your answer -

Answer 1

We are given the distribution here as:

$X \sim U(1,1.5)$

• Clearly as X ranges from 1 to 1.5, therefore the probability is non 0 only for these values of X: 1 < X < 1.5. The first graph is incorrect as it ranges from 0.5 to 1. similarly the second one is also incorrect.
• As the total area under the curve should be 1. Therefore the height of the line is computed as:
  (1.5 - 1)*H = 1
  H = 2

Therefore Graph 3 is the correct graph here.

b) The probability here is computed as:

As the uniform distribution is continuous here, the probability of having a specific value is always 0. Therefore 0 is the required probability here.

c) The probability here is computed as:

$P( 1 \leq x \leq 1.25) = \frac{1.25 - 1}{1.5 - 1} = 0.5$

Therefore 0.5 is the required probability here.

d) The probability here is computed as:

$P( 1.2 < X < 1.5) = \frac{1.5 - 1.2}{1.5 -1} = 0.6$

Therefore 0.6 is the required probability here.