diff --git a/source/main.ptx b/source/main.ptx index 3675fc0..05b796d 100644 --- a/source/main.ptx +++ b/source/main.ptx @@ -34,6 +34,7 @@ Quizzes + diff --git a/source/notes/1-29.ptx b/source/notes/1-29.ptx index 3397a0e..d5cee30 100644 --- a/source/notes/1-29.ptx +++ b/source/notes/1-29.ptx @@ -307,7 +307,7 @@ Let N be the number of heads in n coin flips with bias p. Then N\sim \Bin(n, p). - \E)N) = \sum_{i=0}^n i\cdot b(i; n, p) = \sum_{i = 0}^n i \cdot {n \choose i} p^i (1-p)^{n - i}. + \E(N) = \sum_{i=0}^n i\cdot b(i; n, p) = \sum_{i = 0}^n i \cdot {n \choose i} p^i (1-p)^{n - i}. Gross.

diff --git a/source/quizzes/quiz-02.ptx b/source/quizzes/quiz-02.ptx new file mode 100644 index 0000000..2f0778e --- /dev/null +++ b/source/quizzes/quiz-02.ptx @@ -0,0 +1,205 @@ + + + + + Quiz 2 + + + +

+ The following work should be completed individually. + Use of notes or textbooks is not allowed. + You may use a scientific calculator, not a graphing calculator or phone app. +

+ +

+ Show all work unless instructed otherwise. +

+
+ + + + + + +

+ Write either True or False for each of the following statements. + No justification is required. +

+
+ + + + +

+ Let n, k be integers with 0 \leq k \leq n. + Then {n\choose k} = {n \choose n - k}. +

+
+ + +

+ True. +

+
+
+ + + + +

+ Suppose X is a continuous random variable with pdf f(x) and cdf F(x). + Then \displaystyle{\int_a^b f(x)\ dx = F(b) - F(a)}. +

+
+ + +

+ True. +

+
+
+ + + + +

+ Suppose X is a random variable taking values between 0 and 6. + Then \E(X) = 3. +

+
+ + +

+ False. +

+
+
+
+ + + +

+ Consider the joint distribution for X and Y below. +

+ + + Joint Distribution + + + + + X = 0 + X = 1 + X = 2 + + + + Y = 0 + 0.1 + 0.05 + 0.1 + + + + Y = 1 + 0.3 + 0.15 + 0.3 + + +
+
+ + + + +

+ Find the marginal distributions for X and Y. +

+
+ + +

+ + \Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 + \Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 + \Pr(X = 2) \amp = 0.1 + 0.3 = 0.4 + \Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 + \Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 + +

+
+
+ + + + +

+ Are X and Y independent? +

+
+ + +

+ + \Pr(X = 2, Y = 0) \amp = 0.2 + \Pr(X = 2)\Pr(Y = 0) \amp = (0.4)(0.25) = 0.1 + + Since 0.2 \neq 0.1, X and Y are not independent. +

+
+
+
+
+ + + + + +

+ Suppose a continuous random variable X taking values in [0, 1] has cdf F(x) = 2x^2 - x^4. +

+
+ + + + +

+ Find the pdf f(x). +

+
+ + +

+ + f(x) = F'(x) = 4x - 4x^3. + +

+
+
+ + + + +

+ Find \E(X). +

+
+ + +

+ + \E(X) \amp = \int_0^1 x f(x)\ dx + \amp = \int_0^1 x (4x - 4x^3)\ dx + \amp = \int_0^1 4x^2 - 4x^4\ dx + \amp = \frac{4x^2}{3} - \frac{4x^5}{5}\bigg|_0^1 + \amp = \left(\frac{4}{3} - \frac{4}{5}\right) - (0) + \amp = \frac{8}{15} \approx 0.533. + +

+
+
+
+
+
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