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The two most common expectations are the mean μ X and variance σ X 2 defined by

μ X = E [ X ] = - x f X ( x ) d x
σ X 2 = E [ ( X - μ X ) 2 ] = - ( x - μ X ) 2 f X ( x ) d x .

A very important type of random variable is the Gaussian or normal random variable.A Gaussian random variable has a density function of the following form:

f X ( x ) = 1 2 π σ X exp - 1 2 σ X 2 ( x - μ X ) 2 .

Note that a Gaussian random variable is completely characterized by its mean and variance.This is not necessarily the case for other types of distributions. Sometimes, the notation X N ( μ , σ 2 ) is used to identify X as being Gaussian with mean μ and variance σ 2 .

Samples of a random variable

Suppose some random experiment may be characterized by a random variable X whose distribution is unknown. For example, suppose we are measuring a deterministic quantity v , but our measurement is subject to a random measurement error ε . We can then characterize the observed value, X , as a random variable, X = v + ε .

If the distribution of X does not change over time, we may gain further insight into X by making several independent observations { X 1 , X 2 , , X N } . These observations X i , also known as samples , will be independent random variables and have the same distribution F X ( x ) . In this situation, the X i 's are referred to as i.i.d. , for independent and identically distributed . We also sometimes refer to { X 1 , X 2 , , X N } collectively as a sample, or observation, of size N .

Suppose we want to use our observation { X 1 , X 2 , , X N } to estimate the mean and variance of X . Two estimators which should already be familiar to you are the sample mean and sample variance defined by

μ ^ X = 1 N i = 1 N X i
σ ^ X 2 = 1 N - 1 i = 1 N ( X i - μ ^ X ) 2 .

It is important to realize that these sample estimates are functions of random variables, and are therefore themselves random variables.Therefore we can also talk about the statistical properties of the estimators. For example, we can compute the mean and variance of the sample mean μ ^ X .

E μ ^ X = E 1 N i = 1 N X i = 1 N i = 1 N E X i = μ X
V a r μ ^ X = V a r 1 N i = 1 N X i = 1 N 2 V a r i = 1 N X i = 1 N 2 i = 1 N V a r X i = σ X 2 N

In both [link] and [link] we have used the i.i.d. assumption. We can also show that E [ σ ^ X 2 ] = σ X 2 .

An estimate a ^ for some parameter a which has the property E [ a ^ ] = a is said to be an unbiased estimate. An estimator such that V a r [ a ^ ] 0 as N is said to be consistent . These two properties are highly desirable because they imply that if alarge number of samples are used the estimate will be close to the true parameter.

Suppose X is a Gaussian random variable with mean 0 and variance 1. Use the Matlab function random or randn to generate 1000 samples of X , denoted as X 1 , X 2 , ..., X 1000 . See the online help for the random function . Plot them using the Matlab function plot . We will assume our generated samples are i.i.d.

Write Matlab functions to compute the sample mean and sample variance of [link] and [link] without using the predefined mean and var functions. Use these functions to compute the sample meanand sample variance of the samples you just generated.

Inlab report

  1. Submit the plot of samples of X .
  2. Submit the sample mean and the sample variance that you calculated. Why are they not equal to the true mean and true variance?

Questions & Answers

Calculate price elasticity of demand and comment on the shape of the demand curve of a good ,when its price rises by 20 percentage, quantity demanded falls from 150 units to 120 units.
Helen Reply
5 %fall in price of good x leads to a 10 % rise in its quantity demanded. A 20 % rise in price of good y leads to do a 10 % fall in its quantity demanded. calculate price elasticity of demand of good x and good y. Out of the two goods which one is more elastic.
what is labor
Grace Reply
labor is any physical or mental effort that helps in the production of goods and services
what is profit maximizing level of out put for above hypothetical firm TC = Q3 - 21Q2 + 600 + 1800 P = 600 MC = 3Q2 - 42Q + 600
Sosna Reply
consider two goods X and Y. When the price of Y changes from 10 to 20. The quantity demanded of X changes from 40 to 35. Calculate cross elasticity of demand for X.
sorry it the mistake answer it is question
consider two goods X and Y. When the price of Y changes from 10 to 20. The quantity demanded of X changes from 40 to 35. Calculate cross elasticity of demand for X.
The formula for calculation income elasticity of demand is the percent change in quantity demanded divided by the percent change in income.
what is labor productivity
Lizzy Reply
if the demand function is q=25-4p+p² 1.find elasticity of demand at the point p=5?
Puja Reply
what are some of the difference between monopoly and perfect competition market
Obeng Reply
n a perfectly competitive market, price equals marginal cost and firms earn an economic profit of zero. In a monopoly, the price is set above marginal cost and the firm earns a positive economic profit. Perfect competition produces an equilibrium in which the price and quantity of a good is economic
what are some characteristics of monopoly market
Obeng Reply
explicit cost is seen as a total experiences in the business or the salary (wages) that a firm pay to employee.
Idagu Reply
what is price elasticity
it is the degree of responsiveness to a percentage change in the price of the commodity
economics is known to be the field
John Reply
what is monopoly
Peter Reply
what is taxation
is the compulsory transfer of wealth from the private sector to the public sector
why do monopoly make excess profit in both long run and short run
Adeola Reply
because monopoly have no competitor on the market and they are price makers,therefore,they can easily increase the princes and produce small quantity of goods but still consumers will still buy....
how to identify a perfect market graph
Adeola Reply
what is the investment
investment is a money u used to the business
investment is the purchase of good that are not consumed today but are used in the future to create wealth.
investment is the good that are not consumed
What is supply
 Supply represents how much the market can offer.
it is the quantity of commodity producers produces at the market
what is the effect of scarce resources on producers
Phindu Reply
explain how government taxes and government producer subsidies affect supply
what is economic
Charles Reply
what are the type of economic
macroeconomics,microeconomics,positive economics and negative economics
what are the factors of production
process of production
Basically factors of production are four (4) namely: 1. Entrepreneur 2. Capital 3. Labour and; 4. Land but there has been a new argument to include an addition one to the the numbers to 5 which is "Technology"
what is land as a factor of production
what is Economic
economics is how individuals bussiness and governments make the best decisions to get what they want and how these choices interact in the market
Economics as a social science, which studies human behaviour as a relationship between ends and scarce means, which have alternative uses.
Economics is a science which study human behaviour as a relationship between ends and scarce means
Economics is a social sciences which studies human behavior as a relationship between ends and scarce mean, which have alternative uses.....
types of demand elasticity
Farouq Reply
what is competitive market?
Shantal Reply
a compataive market is when there are many producers competating to provide consumers with a goods and services needed
in a compitative market no single producer or consumer can dictate the market
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
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Source:  OpenStax, Purdue digital signal processing labs (ece 438). OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10593/1.4
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