diff --git a/source/sec-Continuous-RVs.ptx b/source/sec-Continuous-RVs.ptx index 161c649..b19ea2b 100644 --- a/source/sec-Continuous-RVs.ptx +++ b/source/sec-Continuous-RVs.ptx @@ -298,13 +298,42 @@ - +

A continuous random variable X taking values in [1, 4] has p.d.f. f(x) = k(x - \sqrt{x}) for some constant k. - What is the value of k?

-
+ + + + + +

+ What is the value of k? +

+
+ + +

+ k = \frac{6}{17}. +

+
+
+ + + + +

+ Find \Pr(2 \leq X \leq 3). +

+
+ + +

+ \Pr(2 \leq X \leq 3) \approx 0.325 +

+
+
@@ -318,6 +347,12 @@ to find \Pr\left(1 \leq X \leq \frac{3}{2}\right).

+ + +

+ \frac{1}{2}\left(\frac{x^2}{2} - \frac{1}{x}\right) + \frac{1}{4}. \Pr\left(1 \leq X \leq \frac{3}{2}\right) \approx 0.479. +

+
@@ -329,8 +364,29 @@ f(x).

+ + +

+ f(x) = x^2 - 2x. +

+
+ + +

+ Let T \sim \Exp(3). + Find \Pr(1 \leq T \leq 3) and \Pr(T \geq 0.5). +

+
+ + +

+ \Pr(1 \leq T \leq 3) \approx 0.050. \Pr(T \geq 0.5) \approx 0.223. +

+
+
+
\ No newline at end of file diff --git a/source/sec-Discrete-RVs.ptx b/source/sec-Discrete-RVs.ptx index 64fbaba..51c7689 100644 --- a/source/sec-Discrete-RVs.ptx +++ b/source/sec-Discrete-RVs.ptx @@ -398,6 +398,13 @@ Which flip has the highest chance of being the first flip to come up heads?

+ + +

+ \Pr(T = k) = 0.6^{k-1} \cdot 0.4, so every additional flip multiplies the probability by 0.6. + Therefore, the highest value for \Pr(T = k) occurs when k = 1, in which case \Pr(T = 1) = 0.4. +

+