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# Introduction To Probability And Random Variables Pdf

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The idea of a random variable can be confusing. In this video we help you learn what a random variable is, and the difference between discrete and continuous random variables. A discrete probability distribution function has two characteristics:.

The Probability, Random Variables and Estimation Theory course introduces the fundamental statistical tools that are required to analyse and describe advanced signal processing algorithms within the MSc Signal Processing and Communications programme. It provides a unified mathematical framework which is the basis for describing random events and signals, and how to describe key characteristics of random processes. The course covers probability theory, considers the notion of random variables and vectors, how they can be manipulated, and provides an introduction to estimation theory. It is demonstrated that many estimation problems, and therefore signal processing problems, can be reduced to an exercise in either optimisation or integration. While these problems can be solved using deterministic numerical methods, the course introduces Monte Carlo techniques which are the basis of powerful stochastic optimisation and integration algorithms.

## 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable

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. These values are obtained by measuring by a thermometer. Another example of a continuous random variable is the height of a randomly selected high school student. 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 PDF has two characteristics:. A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. 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:. A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a hour shift.

For a random sample of 50 patients, the following information was obtained. Why is this a discrete probability distribution function two reasons? Suppose Nancy has classes three days a week.

She attends classes three days a week 80 percent of the time, two days 15 percent of the time, one day 4 percent of the time, and no days 1 percent of the time. Suppose one week is randomly selected. Describe the random variable in words. Suppose one week is randomly chosen. Construct a probability distribution table called a PDF table like the one in Example 4. The table should have two columns labeled x and P x.

The sum of the P x column is 0. Jeremiah has basketball practice two days a week. Ninety percent of the time, he attends both practices. Eight percent of the time, he attends one practice. Two percent of the time, he does not attend either practice. What is X and what values does it take on? Introduction Introduction There are two types of random variables , discrete random variables and continuous random variables.

A discrete probability distribution function PDF has two characteristics: Each probability is between zero and one, inclusive. The sum of the probabilities is one.

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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.

These ideas are unified in the concept of a random variable which is a numerical summary of random outcomes. Random variables can be discrete or continuous. A basic function to draw random samples from a specified set of elements is the function sample , see? We can use it to simulate the random outcome of a dice roll. The cumulative probability distribution function gives the probability that the random variable is less than or equal to a particular value. For the dice roll, the probability distribution and the cumulative probability distribution are summarized in Table 2.

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.

## Probability density function

Instead, we can usually define the probability density function PDF. The PDF is the density of probability rather than the probability mass. The concept is very similar to mass density in physics: its unit is probability per unit length.

A continuous random variable takes on an uncountably infinite number of possible values. We'll do that using a probability density function "p.

Don't show me this again. This is one of over 2, courses on OCW. Explore materials for this course in the pages linked along the left.

Tentative Grading Scheme. Bunking without Prior Permission from Instructor F :. Bunked is a binary random variable for a student taking on a value of 1 if bunked and 0 if present till mid sem exam.