Mean & Variance of Random Variable

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Mean & Variance of Random Variable (ppt by Aibad Ahmed & Ali Habib of Muhammad Ali Jinnah University)

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Mean and Variance Of R.V:

Mean and Variance Of R.V Probability and Random Signal Ali Habib (Sp10-Bs-0003) Aibad Ahmed (Fa09-Bs-0001)

Table Of Content:

Table Of Content Introduction Mean Properties of Mean Variance Properties of Variance Examples

Introduction:

Introduction What is Random Variable ? “A random variable is a function that associates a unique numerical value with every outcome of an experiment. The value of the random variable will vary from trial to trial as the experiment is repeated”. There are two types of random variable - discrete and continuous. But our topic is related to the Mean and Variance of the random variable

Mean:

Mean The term Mean is used for the process of averaging when a random variable is involved. It is a number used to locate the “center” of the distribution of a random variable. We can also use word “Expectation” for mean. If X is random variable, then the Expectation (expected value or Mean) of X denoted by E[X] is defined by :

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For Continuous Random Variable : For Discrete Random Variable:

Example and Explanation:

Example and Explanation For Continuous random variable

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For Discrete Random Variable

Properties of Mean:

Properties of Mean If a random variable X is adjusted by multiplying by the value b and adding the value a , then the mean is affected as follows The mean of the sum of two random variables X and Y is the sum of their means:

Applications

Variance:

Variance The variance is a measure of the amount of variation of the values of that variable, taking account of all possible values and their probabilities or weightings (not just the extremes which give the range). The variance of a discrete and continuous random variable X measures the spread, or variability, of the distribution, and are defined by

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For Continuous : For Discrete :

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Exponential distribution The exponential distribution with parameter λ is a continuous distribution whose support is the semi-infinite interval [0,∞). Its probability density function is given by : and it has expected value μ = λ −1 .Therefore the variance is equal to: So for an exponentially distributed random variable σ 2 = μ 2 .

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Fair dice A six-sided fair die can be modelled with a discrete random variable with outcomes 1 through 6, each with equal probability . The expected value is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5. Therefore the variance can be computed to be: The general formula for the variance of the outcome X of a die of n sides is:

Properties:

Properties Variance is non-negative because the squares are positive or zero. If all values are scaled by a constant, the variance is scaled by the square of that constant. The variance of a sum of two random variables is given by:

Applications