[PDF] Discrete random variables

Lecture 5: Random variables and expectation

21 តុលា 2020 If X is a simple random variable on a discrete probability space (Ω F

1 Random variables

e.g.: You roll two dice letting X be the random variable referring to the result on the first die

Math 563 - Fall 15 - Homework 3 - selected solutions 4. Prove that a

Prove that a random variable X is independent of itself if and only if there is a constant c such that P(X = c) = 1. Hint: What can you say about the.

Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) = 1/2 x x

] which verifies the third equation. Problem 1.9.7. Show that the moment generating function of the random variable X having the 

POL571 Lecture Notes: Expectation and Functions of Random

10 មីនា 2006 Example 2 Show that b = E(X) minimizes E[(X − b)2]. Finally we emphasize that the independence of random variables implies the mean ...

Feb 21 Homework Solutions Math 151 Winter 2012 Chapter 5

Problem 10. Let f(x) denote the probability density function of a normal random variable with mean. µ and variance σ2. Show that µ − σ and µ + σ are points of 

6 Jointly continuous random variables

And for a continuous random variable X we have a probability density function fX(x). Show that X and Y are independent and find their joint density. Theorem 4 ...

Moment generating functions - 13.1. Definition and examples

Suppose for two random variables X and Y we have mX(t) = mY (t) < ∞ for all t in an interval then X and Y have the same distribution. We will not prove this

Chapter 4: Generating Functions

Practical use: If we can show that two random variables have the same PGF in Theorem 4.4: Let X be a discrete random variable with PGF GX(s). Then: 1. E ...

1 Random variables

So f(x) is a probability distribution on possible outcomes of X(s) which need not be variable group it into categories

Chapter 3 - Random Variables and Measurable Functions.

Definition 43 ( random variable) A random variable X is a measurable func- Proof. For 1. notice that [X1 + X2 > x] if and only if there is a rational.

POL571 Lecture Notes: Expectation and Functions of Random

10 mars 2006 Definition 1 Let X be a random variable and g be any function. ... In particular the following theorem shows that expectation.

1 Subgaussian random variables

X is a bounded and centered random variable with X ? [a

Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) = 1/2 x x

which verifies the third equation. Problem 1.9.7. Show that the moment generating function of the random variable X having the pdf f(x)=1/3 

Some Formulas of Mean and Variance: We consider two random

Theorem: E(XY) = E(X)E(Y) when X is indepen- dent of Y. Proof: For discrete random variables X and Y

Chernoff bounds and some applications 1 Preliminaries

21 févr. 2015 Proof. (of Chebyshev's inequality.) Apply Markov's Inequality to the non-negative random variable (X ?. E(X)). 2. Notice that.

Discrete random variables

For a discrete random variable X we define the probability mass function (PMF) PROOF. Consider a random variable Z := X + Y which is a discrete random ...

Overview 1 Probability spaces

21 mars 2016 Definition A random variable X is a measurable function from a ... Use Hölder's inequality to show that if X is a random variable and q ? 1 ...

Math 710 Homework 5

14 nov. 2010 A random variable X is independent of itself if and only if there is some constant c such that P{X = c} = 1. Proof. (?) Choose some event ? ? ...

Random Variables - MIT - Massachusetts Institute of Technology

Anindicator random variable(or simply anindicatoror aBernoulli random variable) isa random variable that maps every outcome to either 0 or 1 The random variableMisan example If all three coins match thenM= 1; otherwiseM= 0 Indicator random variables are closely related to events

41: Probability Density Functions (PDFs) and Cumulative Distribution

Let X be a discrete rv Then the probability mass function (pmf) f(x) of X is:! f(x)= P(X = x) x ? ? 0 x ? ? Continuous! P(a"X"b)= f(x)dx a b # Let X be a continuous rv Then the probability density function ( pdf ) of X is a function f(x) such that for any two numbers a and b with a ? b: a b A a

Chapter 4 RANDOM VARIABLES - University of Kent

random variable X is the function p(x) satisfying p(x) = Pr(X = x) for all values x in the range of X Abbreviation: pf Notation: p(x) or pX(x) We use the pX(x) form when we need to make the identity of the rv clear Terminology: The pf is sometimes given the alternative name of probability mass function (pmf)

How do you find the PDF of a continuous random variable?

Note that the Fundamental Theorem of Calculus implies that the pdf of a continuous random variable can be found by differentiating the cdf. This relationship between the pdf and cdf for a continuous random variable is incredibly useful. Continuing in the context of Example 4.1.1, we find the corresponding cdf.

Why is X a discrete PDF?

X takes on the values 0, 1, 2, 3, 4, 5. This is a discrete PDF because we can count the number of values of x and also because of the following two reasons: 2 50 + 11 50 + 23 50 + 9 50 + 4 50 + 1 50 = 1 A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift.

What is an indicator random variable?

Anindicator random variable(or simply anindicatoror aBernoulli random variable) is a random variable that maps every outcome to either 0 or 1. The random variable M is an example. If all three coins match, then M = 1; otherwise, M = 0. Indicator random variables are closely related to events.

How do you find the value of a random variable?

The value of this random variable can be 5'2", 6'1", or 5'8". Those values are obtained by measuring by a ruler. A discrete probability distribution function has two characteristics: Each probability is between zero and one, inclusive. The sum of the probabilities is one.