 # Mean and variance of discrete random variable pdf

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## 4.1.2 Expected Value and Variance

Discrete and Continuous Random Variables:.

In probability theory , a probability density function PDF , or density of a continuous random variable , is a function whose value at any given sample or point in the sample space the set of possible values taken by the random variable can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values , as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1. The terms " probability distribution function "  and " probability function "  have also sometimes been used to denote the probability density function.

### Binomial distribution

When introducing the topic of random variables, we noted that the two types — discrete and continuous — require different approaches. The equivalent quantity for a continuous random variable, not surprisingly, involves an integral rather than a sum. Several of the points made when the mean was introduced for discrete random variables apply to the case of continuous random variables, with appropriate modification. Recall that mean is a measure of 'central location' of a random variable. An important consequence of this is that the mean of any symmetric random variable continuous or discrete is always on the axis of symmetry of the distribution; for a continuous random variable, this means the axis of symmetry of the pdf. The module Discrete probability distributions gives formulas for the mean and variance of a linear transformation of a discrete random variable. In this module, we will prove that the same formulas apply for continuous random variables.

The binomial distribution is used to represent the number of events that occurs within n independent trials. Possible values are integers from zero to n. Where equals. In general, you can calculate k! If X has a standard normal distribution, X 2 has a chi-square distribution with one degree of freedom, allowing it to be a commonly used sampling distribution. The sum of n independent X 2 variables where X has a standard normal distribution has a chi-square distribution with n degrees of freedom. The shape of the chi-square distribution depends on the number of degrees of freedom. ## 4.1.2 Expected Value and Variance

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Constructing a probability distribution for random variable. Valid discrete probability distribution examples. Probability with discrete random variable example. Practice: Probability with discrete random variables.

There are two types of random variables , discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. All random variables we discussed in previous examples are discrete random variables. We counted the number of red balls, the number of heads, or the number of female children to get the corresponding random variable values. The values of a continuous random variable are uncountable, which means the values are not obtained by counting. Instead, they are obtained by measuring.

Previous: 2. Next: 2. Analogous to the discrete case, we can define the expected value, variance, and standard deviation of a continuous random variable. These quantities have the same interpretation as in the discrete setting. The expectation of a random variable is a measure of the centre of the distribution, its mean value. The variance and standard deviation are measures of the horizontal spread or dispersion of the random variable. The following animation encapsulates the concepts of the CDF, PDF, expected value, and standard deviation of a normal random variable. With discrete random variables, we often calculated the probability that a trial would result in a particular outcome. For example, we might calculate the probability that a roll of three dice would have a sum of 5. The situation is different for continuous random variables.

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• #### Nesspataho 14.05.2021 at 15:42

• #### Annett D. 16.05.2021 at 10:36

These ideas are unified in the concept of a random variable which is a numerical summary of random outcomes.

• #### Onfroi C. 18.05.2021 at 08:40

Mean and Variance of Discrete Random Variables. Page 2. Expected Value. Variance and Standard Deviation. Practice Exercises. Expected Value of Discrete​.

• #### Laurie C. 19.05.2021 at 09:56 #### Robin A.

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