PDF and CDF define a random variable completely. Standard Deviation ( ) is defined as the square root of variance Properties of Mean and Variance:įor a constant – “c” following properties will hold true for meanįor a constant – “c” following properties will hold true for variance For a continuous random variable X, the variance is defined asįor discrete case, the variance is defined as Variance measures the spread of a distribution. Mean is also called expectation (E)įor continuos random variable X and probability density function f X(x)įor discrete random variable X, the mean is calculated as weighted average of all possible values (x i) weighted with individual probability (p i) Probability of each outcome is used to weight each value when calculating the mean.
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The mean of a random variable is defined as the weighted average of all possible values the random variable can take. Out of these distributions, you will encounter Gaussian distribution or Gaussian Random variable in digital communication very often. This can be easily modeled as a probability density function which will be the integral of probability distribution function with limits 1 to 3.īased on the probability density function or how the PDF graph looks, PDF fall into different categories like binomial distribution, Uniform distribution, Gaussian distribution, Chi-square distribution, Rayleigh distribution, Rician distribution etc. Now the straight forward approach will be to add the probabilities of getting the values which comes out to be. The problem becomes slightly complex if we are asked to find the probability of getting a value less than or equal to 3. The PDF (defined for Continuous Random Variables) is given by taking the first derivate of CDF.įor discrete random variable that takes on discrete values, is it common to defined Probability Mass Function. Its more common deal with Probability Density Function (PDF)/Probability Mass Function (PMF) than CDF. Probability Density function (PDF) and Probability Mass Function(PMF): This equation ( equivalently a function) is called probability distribution function. For example, the equation for this experiment can be given by where. It will be more convenient for us if we have an equation for this experiment which will give these values based on the events. The probabilities of getting the numbers 1,2,3,4 individually are respectively. Probability Distribution function :Ĭonsider an experiment in which the probability of events are as follows. If the values taken by the random variables are of continuous nature (Example: Measurement of temperature), then the random variable is called Continuous Random Variable and the corresponding cumulative distribution function will be smoother without discontinuities. The example provided above is of discrete nature, as the values taken by the random variable are discrete (either “0” or “1”) and therefore the random variable is called Discrete Random Variable. If we plot the CDF for our coin-flipping experiment, it would look like the one shown in the figure on your right. The Cumulative Distribution Function is defined as, Here the bold faced “ X” is a random variable and “x” is a dummy variable which is a place holder for all possible outcomes ( “0” and “1” in the above mentioned coin flipping experiment).
Mathematically, a complete description of a random variable is given be “Cumulative Distribution Function”- F X(x). This means that we can say that the probability of getting Head ( our random variable X = 0 ) as well that of getting Tail ( X =1 ) is 0.5 (i.e. In the coin-flipping experiment, all outcomes are equally probable (given that the coin is fair and unbiased). Because the outcome will lose its significance, we want to associate some probability to each of the possible event. Obviously, we do not want to wait till the coin-flipping experiment is done. This variable “ X” is called a random variable, since it can randomly take any value ‘0’ or ‘1’ before performing the actual experiment. Assign real numbers to the all possible events (this is called “sample space”), say “0” to “Head” and “1” to “Tail”, and associate a variable “ X” that could take these two values. But we know the all possible outcomes – Head or Tail. In a “coin-flipping” experiment, the outcome is not known prior to the experiment, that is we cannot predict it with certainty (non-deterministic/stochastic).
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