lecture_21

Thu Jul 5 09:01:02 CDT 2012

Homework

p.224

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p.227

• 5.7
• 5.8

Practice Problem

Use integration by parts to reduce [\int\limits_{-\infty}^{\infty}{ x^2 e^{-x^2} dx}] to an integral we already know.

• [x^2 e^{-x^2} = \in]

Exam

1a)

[P( A \cup B \cup C) = P( A) + P( B) + P(C) - P( AB) - P( AC) - P( BC)]

1b)

1c)

[\sum\limits_{n = 1}^{\infty} n \left(\frac{ 3}{ 5}\right) \left(\frac{ 2}{ 5}\right)^{n-1}]

[\sum_{n = 0}^{\infty}{ x^n = \frac{ 1}{ 1 -x}]

[\sum_{n = 1}^{\infty} n x^{n - 1} = \frac{ 1}{ (1 - x)^2}]

[\sum_{n = 1}^{\infty}{ n \left(\frac{ 3}{ 5}\right)^{n - 1} = \frac{ 1}{ \left(1 - \frac{ 2}{ 5}\right)^2} = \frac{ 25}{ 9}]

[\frac{ 25}{ 9} \frac{ 3}{ 5}]

Also could have noticed it is an expectation of a gemetric random variable.

3)

[\bigcup\limits_{i = 1}^{3} B_i = S] is a disjoint union

[P( B_1|A) = \frac{ P( B_1 A)}{ P( A)} = \frac{ P( A|B_1) P( B_1)}{ P( A)}]

[P( A) = P( A| B_1)P( B_1) + P( A|B_2)P( B_2) + P( A|B_3) P( B_3)]

so

[P( B_1|A) = \frac{ P( A| B_1) \frac{ 1}{ 2}}{ P( A|B_1) \frac{ 1}{ 2} + P( A|B_2)\frac{ 1}{ 3} + P( A|B_3) \frac{ 1}{6}} = \dots]

4)a)

[\lambda = np = 100 \frac{ 20}{ 100} = 20]

for poisson Distribution Approximation

[E( X) = \lambda = 20]

For Binomial r.v. [(100, \frac{ 1}{ 5})]

[Var( X) = n p(1 - p) = 100 \frac{ 1}{ 5} \frac{ 4}{ 5} = 16]

4) b)

[I = \int\limits_{0}^{\infty} e^{-x^2} dx]

[I^2 = \int\limits_{0}^{\infty} e^{-x^2} dx \int\limits_{0}^{\infty} e^{-y^2} dy]

[I^2 = \int\limits_{0}^{\infty} \int\limits_{0}^{\infty} e^{-(x^2 +y^2)} dx dy]

[ = \int\limits_{0}^{\frac{ \pi}{ 2}} \int\limits_{0}^{\infty} e^{-(r^2)} r dr d \theta = \frac{ \pi}{ 2} \int\limits_{0}^{\infty} r e^{-r^2} dr]

[I = \sqrt{ \frac{ \pi}{ 2} \int\limits_{0}^{\infty} r e^{-r^2} dr}]

7)

[n! \sim \frac{ n^n}{ e^n} \frac{ 1}{ 2 \pi n}]

[\frac{ 1}{ 3^{3n}} \frac{ (3n)!}{ (n!)^3} \sim \frac{ 1}{ 3^{ 3n}}} \frac{ (3n)^{3n}}{ e^{3n}} \sqrt{ 2 \pi 3 n}]

[ \rightarrow \frac{ e^n}{ n^n} \frac{ 1}{ (2 \pi n)^{\frac{ 3}{ 2}}} \sim \frac{ \sqrt{ 2 \pi n} \sqrt{ 3}}{ ( 2 \pi n)^{\frac{ 3}{ 2}}} = \frac{ \sqrt{ 3}}{ 2 \pi n}]