Probability in Continuous Probability Distribution

In a continuous probability distribution, the probability that a specific value (x) assumes a specified value is always zero. This is because in a continuous distribution, the possible values of the random variable are uncountable and form a continuum.

Continuous Probability Distribution
A continuous probability distribution refers to a probability distribution where the random variable can take on any value within a given range, usually represented by an interval. Examples of continuous probability distributions include the normal distribution, exponential distribution, and uniform distribution.

Probability Density Function (PDF)
In a continuous probability distribution, the probability of a random variable taking on a specific value is defined by the probability density function (PDF). The PDF represents the relative likelihood of the random variable assuming a particular value within the given range.

Area Under the Curve
In a continuous probability distribution, the probability of an event occurring is determined by the area under the curve of the PDF. The total area under the curve represents the probability of the entire range of possible values, which is equal to 1.

Probability of a Single Value
Since the probability is determined by the area under the curve, the probability of a single specific value occurring is essentially zero. This is because the area of a single point on a continuous curve is infinitesimally small.

Interval Probability
In a continuous probability distribution, the probability is typically defined over an interval rather than a single value. The probability of the random variable falling within a specific interval can be calculated by finding the area under the curve within that interval.

Conclusion
In summary, the probability that a specific value assumes a specified value in a continuous probability distribution is always zero. This is because the probability is determined by the area under the curve, and a single point on a continuous curve has infinitesimally small area.