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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.
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The integral of the standard normal distribution function is an integral without solution and represents the probability that an aleatory variable normally distributed has values between zero and. The normal distribution integral is used in several areas of science. Thus, this work provides an approximate solution to the Gaussian distribution integral by using the homotopy perturbation method HPM.
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ReplyUsing this for the function u:x↦Φ((x−b)/a) and assuming furthermore that X and Y are independent, one sees that the integral to be computed is (∗)=E(Φ((X−a)/b).
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