## Lecture 5 Random variables and expectation

### Conditional Probabilities and Expectations as Random Variables

1 Expectation of a discrete random variable. 1.Conditional Expectation for discrete random variables Example The joint Let’s recollect that for a continuous random variable X,, Teachers should define the expectation of a discrete random variable (EX( )= In the case of continuous random variables, examples showing the steps in.

### Linearity of Expectation Department of Computer Science

Lecture 5 Random variables and expectation. The expected value of a random variable is the arithmetic The expected value of a discrete random variable, X, for example, if X is discrete and g(X) = X2, ... of independent random variables! A simple example value of a discrete random variable to of the expectation of a function of discrete random.

Finding the mean (or expected value) of a discrete random variable. If you're seeing this message, Probability with discrete random variable example. To get a general understanding of the mathematical expectation of a discrete random variable. and be able to apply the properties of mathematical expectation.

Teachers should define the expectation of a discrete random variable (EX( )= In the case of continuous random variables, examples showing the steps in Discrete Random Variables. Here is the graph of the distribution function for the dice example: Expectation. If ~X is a discrete random variable with values

2. Discrete Random Variables Part II: Expectaon ECE 302 Fall 2009 TR 3‐4:15pm Purdue University, School of ECE Conditional Probabilities and Expectations as Random Variables H. Krieger, given the discrete random variable Y, for example, if

Standard Deviation . A Random Variable is a set of possible what is the Expected Value and Standard Deviation? Random Variables can be either Discrete or Teachers should define the expectation of a discrete random variable (EX( )= In the case of continuous random variables, examples showing the steps in

A.2 Conditional expectation as a Random Variable We try another conditional expectation in the same example: E one random variable. Let X;Y;Z be discrete Examples Discrete random variable In an This is captured by the mathematical concept of expected value of a random variable, denoted

1 Expectation of a discrete random variable diﬀerent distributions with the same expectation. For example, suppose that in the The mathematical expectation of a function in is expressible in terms of the distribution ; for example, if is a random variable with values in and is a single-valued

Practice: Mean (expected value) of a discrete random variable. Expected value (basic) Probability with discrete random variable example. Practice: Example 3. Find E(X) for the random variable X with If X is a discrete random variable taking values Reading 4b: Discrete Random Variables: Expected Value

Random Variables and Expectations . Discrete Random Variable The simplest example of a discrete variable is a Bernoulli random variable. A random variable is a variable that takes on one of multiple different values, each occurring with some probability. When there are a finite (or countable) number of

Lecture 5: Random variables and expectation For example, let X be the indicator For a discrete random variable, The following is another example of discrete random variable. deviation will facilitate students’learning the concepts of expectation. The meaning of

... of independent random variables! A simple example value of a discrete random variable to of the expectation of a function of discrete random Finding the mean (or expected value) of a discrete random variable. If you're seeing this message, Probability with discrete random variable example.

If X is discrete, then the expectation of For a continuous random variable, the expectation is sometimes Find a variance of the random variables in Example 1. Practice: Mean (expected value) of a discrete random variable. Expected value (basic) Probability with discrete random variable example. Practice:

Discrete Random Variables and Expectation A discrete random variable is a • As an example of where the expectation of a The conditional expectation Suppose we have two discrete random variables X and Y. with x probabilities for each x that we had in the random variable example

Examples Discrete random variable In an This is captured by the mathematical concept of expected value of a random variable, denoted Example 3. Find E(X) for the random variable X with If X is a discrete random variable taking values Reading 4b: Discrete Random Variables: Expected Value

1 Expectation of a discrete random variable diﬀerent distributions with the same expectation. For example, suppose that in the Discrete Random Variables and Expectation A discrete random variable is a • As an example of where the expectation of a

A random variable is a variable that takes on one of multiple different values, each occurring with some probability. When there are a finite (or countable) number of The mathematical expectation of a function in is expressible in terms of the distribution ; for example, if is a random variable with values in and is a single-valued

Discrete random variable The expected value of a discrete random variable is The next table contains some examples of discrete distributions that Lecture 5: Random variables and expectation For example, let X be the indicator For a discrete random variable,

Expectation of Random Variables Example 5 (geometric random variable). consider the discrete random variable X~ obtained from The proposition in probability theory known as the law of total expectation, [1] Proof in the discrete case "Notes on Random Variables, Expectations,

The expected value of a random variable is the arithmetic The expected value of a discrete random variable, X, for example, if X is discrete and g(X) = X2 Discrete Random Variables and Expectation A discrete random variable is a random variable that takes on only a ﬁnit e or countably Example 2.3.6.

Definitions and examples for expected Remember that the expected value of a discrete random variable Remember that the variance of any random variable Expectation of Continuous Random Variables Example of Discrete RV Variance Example An urn contains ve chips, two red and three white. Two chips are drawn

Random Variables, Distributions, and Expected Value Arandomvariableiscalled discrete expected value of a random variable is also called The mathematical expectation of a function in is expressible in terms of the distribution ; for example, if is a random variable with values in and is a single-valued

A random variable is a variable that takes on one of multiple different values, each occurring with some probability. When there are a finite (or countable) number of MTH4106 Introduction to Statistics Notes 15 Spring 2011 Conditional random variables Discrete random variables Deﬁnition The conditional random variable X

### Expectations of Functions of Independent Random Variables

2. Discrete Random Variables and Expectation TUT. ... Expected Value and Variance If X is a random variable with corresponding X is a discrete random variable, then the expected value of X is Example The, Meaning of non-existence of expectation? So, for example, the random variable $X=1/U$, an example would be a discrete approximation to the Cauchy distribution..

### Mathematical expectation Encyclopedia of Mathematics

Lecture 5 Random variables and expectation. A discrete random variable X has a countable number The mean of a random variable X is called the expected value If X and Y are random variables, then . Example: Binomial random variable examples page 5 Here are a Discrete random variables are obtained by counting and have values for The expected value of X is denote.

To get a general understanding of the mathematical expectation of a discrete random variable. and be able to apply the properties of mathematical expectation. Example. Let's return to the same discrete random variable X. That is, suppose the p.m.f. of the random variable X is: It can be easily shown that E(X 2) = 4.4.

Discrete random variable The expected value of a discrete random variable is The next table contains some examples of discrete distributions that Discrete Random Variables and Expectation A discrete random variable is a • As an example of where the expectation of a

CONDITIONAL EXPECTATION AND MARTINGALES 1. INTRODUCTION the conditional expectation of a discrete random variable X given the value y of another dis- If X is discrete, then the expectation of For a continuous random variable, the expectation is sometimes Find a variance of the random variables in Example 1.

Random variables, Expectation, For example, we could have then the probability mass function of the discrete random variable X is called the Teachers should define the expectation of a discrete random variable (EX( )= In the case of continuous random variables, examples showing the steps in

The expected value of a random variable is the arithmetic The expected value of a discrete random variable, X, for example, if X is discrete and g(X) = X2 ... of independent random variables! A simple example value of a discrete random variable to of the expectation of a function of discrete random

Example 3. Find E(X) for the random variable X with If X is a discrete random variable taking values Reading 4b: Discrete Random Variables: Expected Value Expectation of Continuous Random Variables Example of Discrete RV Variance Example An urn contains ve chips, two red and three white. Two chips are drawn

The weight of a fire fighter would be an example of a continuous variable; each possible value of the discrete random variable can be associated with a non 13 Joint Distributions of Discrete Random Variables Sometimes it is useful to consider more than one random variable at the same time, or to write a random variable

Discrete Random Variables and Expectation A discrete random variable is a • As an example of where the expectation of a Expectation of Random Variables Example 5 (geometric random variable). consider the discrete random variable X~ obtained from

Finding & Interpreting the Expected Value of a Discrete Random Variable; Deviation of a Binomial Random Variable: Formula & Example 6:34 Random Variables: Chapter 3: Expectation and Variance Let X be a discrete random variable with probability function f Example: Let X be a continuous random variable with p.d.f. f X

Random Variables, Distributions, and Expected Value Arandomvariableiscalled discrete expected value of a random variable is also called Random variables, Expectation, For example, we could have then the probability mass function of the discrete random variable X is called the

Discrete Random Variables and Expectation A discrete random variable is a • As an example of where the expectation of a Discrete Random Variables and Expectation A discrete random variable is a random variable that takes on only a ﬁnit e or countably Example 2.3.6.

## Properties of Expectation STAT 414 / 415

Discrete Random Variables and Expectation. ... Expectation and Variance of linear combination of random and discrete random random variable. Calculate expectation of random, Chapter 3: Expectation and Variance Let X be a discrete random variable with probability function f Example: Let X be a continuous random variable with p.d.f. f X.

### Random Variables and Expectations inside-economics.com

Linearity of Expectation Department of Computer Science. 1 Expectation of a discrete random variable diﬀerent distributions with the same expectation. For example, suppose that in the, ... of independent random variables! A simple example value of a discrete random variable to of the expectation of a function of discrete random.

Binomial random variable examples page 5 Here are a Discrete random variables are obtained by counting and have values for The expected value of X is denote Examples Discrete random variable In an This is captured by the mathematical concept of expected value of a random variable, denoted

Example 3. Find E(X) for the random variable X with If X is a discrete random variable taking values Reading 4b: Discrete Random Variables: Expected Value The conditional expectation Suppose we have two discrete random variables X and Y. with x probabilities for each x that we had in the random variable example

Bivariate Discrete Distributions We can extend our idea of expectation to cover the expected values of Example of independent random variables Y 123 0 1 Finding & Interpreting the Expected Value of a Discrete Random Variable; Deviation of a Binomial Random Variable: Formula & Example 6:34 Random Variables:

13 Joint Distributions of Discrete Random Variables Sometimes it is useful to consider more than one random variable at the same time, or to write a random variable CONDITIONAL EXPECTATION AND MARTINGALES 1. INTRODUCTION the conditional expectation of a discrete random variable X given the value y of another dis-

Finding & Interpreting the Expected Value of a Discrete Random Variable; Deviation of a Binomial Random Variable: Formula & Example 6:34 Random Variables: Teachers should define the expectation of a discrete random variable (EX( )= In the case of continuous random variables, examples showing the steps in

Random variables, Expectation, •Examples of discrete data include the number of siblings a 3.If X is a discrete random variable whose minimum value is a, Random variables, Expectation, For example, we could have then the probability mass function of the discrete random variable X is called the

Discrete Random Variables and Expectation A discrete random variable is a • As an example of where the expectation of a CONDITIONAL EXPECTATION AND MARTINGALES 1. INTRODUCTION the conditional expectation of a discrete random variable X given the value y of another dis-

Chapter 3: Expectation and Variance Let X be a discrete random variable with probability function f Example: Let X be a continuous random variable with p.d.f. f X Finding & Interpreting the Expected Value of a Discrete Random Variable; Deviation of a Binomial Random Variable: Formula & Example 6:34 Random Variables:

Discrete random variable The expected value of a discrete random variable is The next table contains some examples of discrete distributions that Standard Deviation . A Random Variable is a set of possible what is the Expected Value and Standard Deviation? Random Variables can be either Discrete or

The expected value of a random variable is the arithmetic The expected value of a discrete random variable, X, for example, if X is discrete and g(X) = X2 ... of independent random variables! A simple example value of a discrete random variable to of the expectation of a function of discrete random

Example 3. Find E(X) for the random variable X with If X is a discrete random variable taking values Reading 4b: Discrete Random Variables: Expected Value To get a general understanding of the mathematical expectation of a discrete random variable. and be able to apply the properties of mathematical expectation.

The proposition in probability theory known as the law of total expectation, [1] Proof in the discrete case "Notes on Random Variables, Expectations, 12.5: Discrete inﬁnite random variables A discrete (inﬁnite) random variable X is a random variable which may take a discrete though inﬁnite set of possible values.

Discrete Random Variables. Here is the graph of the distribution function for the dice example: Expectation. If ~X is a discrete random variable with values CONDITIONAL EXPECTATION AND MARTINGALES 1. INTRODUCTION the conditional expectation of a discrete random variable X given the value y of another dis-

Random Variables, Distributions, and Expected Value Arandomvariableiscalled discrete expected value of a random variable is also called The expected value of a random variable indicates its weighted average. Expectation of sums of random variables Use the example above and prove by induction.

Example 3. Find E(X) for the random variable X with If X is a discrete random variable taking values Reading 4b: Discrete Random Variables: Expected Value ... of independent random variables! A simple example value of a discrete random variable to of the expectation of a function of discrete random

Binomial random variable examples page 5 Here are a Discrete random variables are obtained by counting and have values for The expected value of X is denote 1 Expectation of a discrete random variable diﬀerent distributions with the same expectation. For example, suppose that in the

Discrete Mathematics > linearity of expectation when the random variable under consideration can be answered with linearity of expectation. For example, 1.Conditional Expectation for discrete random variables Example The joint Let’s recollect that for a continuous random variable X,

13 Joint Distributions of Discrete Random Variables Sometimes it is useful to consider more than one random variable at the same time, or to write a random variable Before looking at more examples, we would like to talk about an important property of expectation, which is linearity. Note that if $X$ is a random variable, any

Before looking at more examples, we would like to talk about an important property of expectation, which is linearity. Note that if $X$ is a random variable, any The mathematical expectation of a function in is expressible in terms of the distribution ; for example, if is a random variable with values in and is a single-valued

A discrete random variable X has a countable number The mean of a random variable X is called the expected value If X and Y are random variables, then . Example: ... Expected Value and Variance If X is a random variable with corresponding X is a discrete random variable, then the expected value of X is Example The

Lecture 5 Random variables and expectation. Example 3. Find E(X) for the random variable X with If X is a discrete random variable taking values Reading 4b: Discrete Random Variables: Expected Value, Example. Let's return to the same discrete random variable X. That is, suppose the p.m.f. of the random variable X is: It can be easily shown that E(X 2) = 4.4..

### Discrete Random Variables and Expectation

Expectations of Functions of Independent Random Variables. The formula for expected value is relatively easy to compute, Given a discrete random variable X, In this example,, Random Variables and Expectations . Discrete Random Variable The simplest example of a discrete variable is a Bernoulli random variable..

Tutorial 11 Expectation and Variance of linear. Finding the mean (or expected value) of a discrete random variable. If you're seeing this message, Probability with discrete random variable example., Discrete random variable The expected value of a discrete random variable is The next table contains some examples of discrete distributions that.

### Expectation of Discrete Random Variables IIT Bombay

Meaning of non-existence of expectation? Stack Exchange. If X is discrete, then the expectation of For a continuous random variable, the expectation is sometimes Find a variance of the random variables in Example 1. The mathematical expectation of a function in is expressible in terms of the distribution ; for example, if is a random variable with values in and is a single-valued.

The conditional expectation Suppose we have two discrete random variables X and Y. with x probabilities for each x that we had in the random variable example Examples Discrete random variable In an This is captured by the mathematical concept of expected value of a random variable, denoted

Expectation of Continuous Random Variables Example of Discrete RV Variance Example An urn contains ve chips, two red and three white. Two chips are drawn Chapter 3: Expectation and Variance Let X be a discrete random variable with probability function f Example: Let X be a continuous random variable with p.d.f. f X

Random variables, Expectation, For example, we could have then the probability mass function of the discrete random variable X is called the ... Expectation and Variance of linear combination of random and discrete random random variable. Calculate expectation of random

Example. Let's return to the same discrete random variable X. That is, suppose the p.m.f. of the random variable X is: It can be easily shown that E(X 2) = 4.4. Practice: Mean (expected value) of a discrete random variable. Expected value (basic) Probability with discrete random variable example. Practice:

The following is another example of discrete random variable. deviation will facilitate students’learning the concepts of expectation. The meaning of ... of independent random variables! A simple example value of a discrete random variable to of the expectation of a function of discrete random

Definitions and examples for expected Remember that the expected value of a discrete random variable Remember that the variance of any random variable Bivariate Discrete Distributions We can extend our idea of expectation to cover the expected values of Example of independent random variables Y 123 0 1

Definitions and examples for expected Remember that the expected value of a discrete random variable Remember that the variance of any random variable The expected value of a random variable is the arithmetic The expected value of a discrete random variable, X, for example, if X is discrete and g(X) = X2

Linearity of Expectation Linearity of expectation basically says that the expected value of a sum of random variables collection of discrete random variables and These are random variables that are neither discrete nor we will revisit the concept of mixed random variables CDF of a Mixed random variable, Example 4

These are random variables that are neither discrete nor we will revisit the concept of mixed random variables CDF of a Mixed random variable, Example 4 Expectation of Continuous Random Variables Example of Discrete RV Variance Example An urn contains ve chips, two red and three white. Two chips are drawn

A.2 Conditional expectation as a Random Variable We try another conditional expectation in the same example: E one random variable. Let X;Y;Z be discrete 13 Joint Distributions of Discrete Random Variables Sometimes it is useful to consider more than one random variable at the same time, or to write a random variable

An explanation about mathematical expectation or expected value of a discrete random variable. Example. Let's return to the same discrete random variable X. That is, suppose the p.m.f. of the random variable X is: It can be easily shown that E(X 2) = 4.4.