# continuous random variable

## continuous random variable

A continuous random variable is a random variable where the data can take infinitely many values. This means that we must calculate a probability for a continuous random variable over an interval and not for any particular point. It is always in the form of an interval, and the interval may be very small. A normal random variable is drawn from the classic "bell curve," the distribution: f(x)=12πσ2e−(x−μ)22σ2,f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}},f(x)=2πσ2​1​e−2σ2(x−μ)2​. Solution: (2) The possible sets of outcomes from flipping ten coins. {\displaystyle X} is called a continuous random variable. (1) The sum of numbers on a pair of two dice. The quantity $$f\left( x \right)\,dx$$ is called probability differential. For example, a random variable measuring the time taken for something to be done is continuous since there are an infinite number of possible times that can be taken. If $$c \geqslant 0$$, $$f\left( x \right)$$ is clearly $$\geqslant 0$$ for every x in the given interval. ), giving c = 2. The computer time (in seconds) required to process a certain program. (4) and (5) are the continuous random variables. Sign up, Existing user? 2. for every subset I ⊂ R, P(X ∈ I) = Z Recall that a random variable is a quantity which is drawn from a statistical distribution, i.e. It is denoted by $$f\left( x \right)$$ where $$f\left( x \right)$$ is the probability that the random variable $$X$$ takes the value between $$x$$ and $$x + \Delta x$$ where $$\Delta x$$ is a very small change in $$X$$. Find c. If we integrate f(x) between 0 and 1 we get c/2. is found by integrating the p.d.f. For example, a random variable measuring the time taken for something to be done is continuous since there are an infinite number of possible times that can be taken. : As with discrete random variables, Var(X) = E(X2) - [E(X)]2. (5) The possible times that a person arrives at a restaurant. If X is a continuous random variable with p.d.f. Baseball batting averages, IQ scores, the length of time a long distance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores are just a few. A continuous random variable is a random variable where the data can take infinitely many values. The number of possible outcomes of a continuous random variable is uncountable and infinite. Review • Continuous random variable: A random variable that can take any value on an interval of R. • Distribution: A density function f: R → R+ such that 1. non-negative, i.e., f(x) ≥ 0 for all x. Formally: A continuous random variable is a function XXX on the outcomes of some probabilistic experiment which takes values in a continuous set VVV. In particular, quantum mechanical systems often make use of continuous random variables, since physical properties in these cases might not even have definite values. In the next article on continuous probability density functions, the meaning of XXX will be explored in a more practical setting. (3) This case is more interesting because there are infinitely many coins. Continuous random variables describe outcomes in probabilistic situations where the possible values some quantity can take form a continuum, which is often (but not always) the entire set of real numbers R\mathbb{R}R. They are the generalization of discrete random variables to uncountably infinite sets of possible outcomes. The peak of the normal distribution is centered at μ\muμ and σ2\sigma^2σ2 characterizes the width of the peak. The formula for the expected value of a continuous random variable is the continuous analog of the expected value of a discrete random variable, where instead of summing over all possible values we integrate (recall Sections 3.6 & 3.7).

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