Probability & Statistics Ch.2 Section 4 Question 2

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

$W \sim \text{Uniform}[1, 4]$. Compute each of the following:

• a) $P(W \geq 5)$
• b) $P(W \geq 2)$
• c) $P(W^2 \geq 9)$
• d) $P(W^2 \leq 2)$

Solution

From Example 2.4.3 in the textbook:

In this case R = 4, L = 1, so the density is $\frac{1}{3}$ on [1,4] and 0 elsewhere.

(a) $P(W \geq 5)$

Base = intersection of $[5, \infty) \cap [1,4] = \emptyset$

Since none of our function’s domain intersects with the uniform distribution domain:

(b) $P(W \geq 2)$

Base = intersection of $[2, \infty) \cap [1,4] = [2,4]$, length = 2

Height = $\frac{1}{4-1} = \frac{1}{3}$

(c) $P(W^2 \geq 9)$

If $W^2 \geq 9$, then $W \geq 3$ or $W \leq -3$.

Base = intersection of $((-\infty,-3] \cup [3,\infty)) \cap [1,4] = [3,4]$, length = 1

Height = $\frac{1}{3}$

(d) $P(W^2 \leq 2)$

If $W^2 \leq 2$, then $-\sqrt{2} \leq W \leq \sqrt{2}$.

Base = intersection of $[-\sqrt{2}, \sqrt{2}] \cap [1,4] = [1, \sqrt{2}]$

Length = $\sqrt{2} - 1 \approx 0.4142$

Height = $\frac{1}{3}$