# Continuous Probability Distribution Examples And Solutions Pdf

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- Probability Distributions: Discrete and Continuous
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- Continuous and discrete probability distributions

A continuous distribution describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable with a set of possible values known as the range that is infinite and uncountable. Probabilities of continuous random variables X are defined as the area under the curve of its PDF. Thus, only ranges of values can have a nonzero probability. The probability that a continuous random variable equals some value is always zero.

## Probability Distributions: Discrete and Continuous

All probability distributions can be classified as discrete probability distributions or as continuous probability distributions, depending on whether they define probabilities associated with discrete variables or continuous variables. If a variable can take on any value between two specified values, it is called a continuous variable ; otherwise, it is called a discrete variable. Just like variables, probability distributions can be classified as discrete or continuous.

If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution. An example will make this clear.

Suppose you flip a coin two times. Now, let the random variable X represent the number of Heads that result from this experiment. The random variable X can only take on the values 0, 1, or 2, so it is a discrete random variable. The above table represents a discrete probability distribution because it relates each value of a discrete random variable with its probability of occurrence.

On this website, we will cover the following discrete probability distributions. Note: With a discrete probability distribution, each possible value of the discrete random variable can be associated with a non-zero probability.

Thus, a discrete probability distribution can always be presented in tabular form. If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution. A continuous probability distribution differs from a discrete probability distribution in several ways. Most often, the equation used to describe a continuous probability distribution is called a probability density function.

Sometimes, it is referred to as a density function , a PDF , or a pdf. For a continuous probability distribution, the density function has the following properties:. For example, consider the probability density function shown in the graph below.

Suppose we wanted to know the probability that the random variable X was less than or equal to a. The probability that X is less than or equal to a is equal to the area under the curve bounded by a and minus infinity - as indicated by the shaded area.

Note: The shaded area in the graph represents the probability that the random variable X is less than or equal to a. This is a cumulative probability. However, the probability that X is exactly equal to a would be zero. A continuous random variable can take on an infinite number of values. The probability that it will equal a specific value such as a is always zero. Probability Probability Basics About the tutorial What is probability? Sets and subsets Statistical experiment Probability Problems Rules of probability Counting data points Probability problems Venn diagrams Bayes theorem Poker Probability Probability in stud poker Probability of straight Probability of flush Cards of equal rank Probability of no pair Random Variables What is a random variable?

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A continuous random variable takes on an uncountably infinite number of possible values. We'll do that using a probability density function "p. We'll first motivate a p. Even though a fast-food chain might advertise a hamburger as weighing a quarter-pound, you can well imagine that it is not exactly 0. One randomly selected hamburger might weigh 0. What is the probability that a randomly selected hamburger weighs between 0. In reality, I'm not particularly interested in using this example just so that you'll know whether or not you've been ripped off the next time you order a hamburger!

The graph of a continuous probability distribution is a curve. Probability is represented by area under the curve. We have already met this concept when we developed relative frequencies with histograms in Chapter 2. The relative area for a range of values was the probability of drawing at random an observation in that group. Again with the Poisson distribution in Chapter 4 , the graph in Example 4. In this case, we were being a bit casual because the random variables of a Poisson distribution are discrete, whole numbers, and a box has width. Notice that the horizontal axis, the random variable x, purposefully did not mark the points along the axis.

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 " [3] and " probability function " [4] have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function , or it may be a probability mass function PMF rather than the density.

## Continuous and discrete probability distributions

Random Variables play a vital role in probability distributions and also serve as the base for Probability distributions. Before we start I would highly recommend you to go through the blog — understanding of random variables for understanding the basics. Today, this blog post will help you to get the basics and need of probability distributions. What is Probability Distribution?

Say you were weighing something, and the random variable is the weight. Even if you could give a probability for, say, Between each two rational numbers there is another one, and so on and so on.

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.

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## 5 Comments

Paulino Q.understand the use of continuous probability distributions and the use of Example. Find the median of this p.d.f.. f x() = x. 4., 1< x < 3. Solution. Now x. 4. 1 m.

Othello P.temperature are continuous, in practice the limitations of or probability density function (pdf) of X is a function f(x) The pdf and probability from Example 4.

RiapesoumoThe common practice in such cases is to say that the possible The probability density function (pdf) f (x) of a continuous random variable X is de- fined as the.

Brittany M.Because there are infinite values that X could assume, the probability of X taking on any one specific value is zero.

Evrard L.Problem. Let X be a continuous random variable with PDF given by fX(x)=12e−|x|,for all x∈R. If Y=X2, find the CDF of Y. Solution. First, we note that RY=[0,∞).