Lecture 3 Continuous Random Variable - 國立中興大學 · Lecture 3 Continuous Random Variable...

Lecture 3

Continuous Random Variable

2012/10/1 2010 Fall Random Process EE@NCHU

Cumulative Distribution Function

Definition

Theorem 3.1 For any random variable X,

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Continuous Random Variable

Definition


Example Suppose we have a wheel of circumference one meter and we mark a

point on the perimeter at the top of the wheel. In the center of the wheel is a radial pointer that we spin. After spinning the pointer, we measure the distance, X meters, around the circumference of the wheel going clockwise from the marked point to the pointer position as shown in the following figure. Clearly, 0 ≤ X < 1. Also, it is reasonable to believe that if the spin is hard enough, the pointer is just as likely to arrive at any part of the circle as at any other. (Note that the random pointer on disk of circumference 1)

(a) For a given x, what is the probability P[X = x]? (b) What is the CDF of X?

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Circumference: 周長

[Continued]

Solution

a) A reasonable approach is to find a discrete approximation to X. Marking the perimeter with n equal-length arcs numbered 1 to n. Let Y

denote the number of the arc in which the pointer stops. Denote the range of Y by SY = {1, 2,…, n}. The PMF of Y is

From the wheel on the right side of the figure, we can deduce that if X = x, then Y =

, where the notation is defined as ’s ceiling integer


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數學符號

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Solution (con’t)

=> The is true regardless of the outcome, x. It follows that every outcome has probability ZERO.

imply

How to find P[X = x] ?

This demonstrates that P[X = x] ≤ 0.

P[X = x] ≥0. Therefore, P[X = x] = 0.

Solution (con’t)


How to find the CDF for X, SX = [0, 1)?

FX(x) = 0 for x < 0 FX(x) = 1 for x ≥ 1.

Solution (con’t)


x

Quiz


Probability Density Function

Definition


slope


Properties of PDF

Theorem 3.2


Proof


Theorem 3.1

Theorem

Proof:


Example


Solution


Quiz


Expected Value of Continuous RV


Definition

Problem

We found that the stopping point X of the spinning wheel experiment was a uniform random variable with PDF

Find the expected stopping point E[X] of the pointer.

Solution:


Expected Value of g(X)

Properties (Theorem 3.5) For any random variable X,


Theorem 3.4

Fine the variance and standard deviation of the pointer position X in the wheel example.

Example


Solution


Quiz

( )kk E Xµ =

( )

( )-

if is discrete

if is continuous

k

x

k

x p x X

x f x dx X∞

∞

=

∑

∫

The kth moment of X

Families of continuous Random Variables

Uniform

Exponential

Erlang

Gaussian

Standard Normal

Standard Normal Complement


Uniform Random Variable

Definition


Properties of Uniform


Theorem 3.6

Problem


Solution


Property of Uniform


Theorem 3.7

Proof

Definition 2.9 Discrete Uniform


f

Exponential Random Variable


Definition

Properties of Exponential


Theorem 3.8

Example

(a) What is the PDF of the duration in minutes of a telephone conversation? (b) What is the probability that a conversation will last between 2 and 4

minutes? (c) What is the expected duration of a telephone call? (d) What are the variance and standard deviation of T? (e) What is the probability that a call duration is within 1 standard

deviation of the expected call duration?


Solution (a)

(b)

[Continued]


Solution (con’t)

(c)

(d)


Solution (con’t)

(e)


X is an exponential (λ) random variable

More Properties of Exponential


Theorem 3.9

Proof

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2010 Fall Random Process EE@NCHU

Example


Solution


Since E [T ] = 1/λ = 3, for company B, which charges for the exact duration of a call,

Because is a geometric random variable with , therefore, the expected revenue for Company A is

Definition order

Note 1: Erlang distribution is the distribution of the sum of k iid exponential random variables. The rate of Erlang distribution is the rate of this exponential distribution. Erlang(1, λ) is identical to Exponential(λ).

Note 2: Erlang is also related to Pascal.

Erlang (n,λ) Random Variable

X is an exponential (λ) random variable

Poisson (α=λT)

Properties of Erlang


Theorem 3.10

Theorem 3.11

Proof (Theorem 3.11)

Proof (Theorem 3.11) (con’t)

Quiz


Gaussian Random Variable

55

Definition

Property of Gaussian


Theorem 3.13

Theorem 3.12

If X is a Gaussian (μ,σ) random variable,

Definition

Standard Normal Random Variable

The standard normal random variable Z is the Gaussian (0,1) random variable. The CDF of the standard normal random variable Z is


Standard Normal Random Variable


Theorem 3.14

Example

Solution:


Symmetry property of Gaussian

Symmetry properties of standard normal or Gaussian (0,1) PDF.

Theorem 3.15

Q

Standard Normal Complement CDF

Definition

Example

Solution:


Quiz


Definition

Unit Impulse (Delta) Function


Property of Unit Impulse


Theorem 3.16 shifting property

Unit Step Function

, where u(x) is the unit step function, defined as


Theorem 3.17

Definition

Example


Remark: Using the Dirac delta function we can define the density function for a discrete random variables.

Example (con’t)

PMF CDF PDF

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From the example, we can see that the discrete random variable Y can either represented by a PMF PY(y) with bar at y = 1,2,3, by a CDF with jumps at y = 1,2,3, or by a PDF fY(y) with impulses at y =1,2,3.

The expected value of Y can be caluculate either by summing over the PMF PY(y) or integrating over the PDF fY(y). Using PDF, we have

Equivalent Statements


Theorem 3.18

Mixed Random Variable


Definition

Example


Solution


Solution (con’t)


Example


Solution


Solution (con’t)


Example


Solution


Solution (con’t)


Quiz


Probability Models of g(X)

Proof:


Theorem 3.19

Example

Solution:


Example

Y centimeters is the location of the pointer on the 1-meter circumference

of the circle. Note that X is the location of the pointer in a unit of meter . Use X to derive fY(y).


Solution

The function Y=100X. To find the PDF of Y, we first find the CDF FY(y). Remind that the CDF of X is


Distributions of Y=g(X)


Theorem 3.20

Property of Derived RV --Shift


Theorem 3.21

Proof:

Theorem 3.22

Proof:


Conditional PDF given an Event

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Definition

Conditional PDF


Definition

Conditional Expected Value given and Event


Definition

Problem

Solution:

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Solution (con’t)


Problem


Solution


Quiz


Lecture 3 Continuous Random Variable - 國立中興大學 · Lecture 3 Continuous Random Variable...

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