Final answer:
The cumulative distribution function (CDF) for a continuous random variable is defined as P(X ≤ x) and represents the probability of the variable being less than or equal to a certain value. For continuous distributions, it can also be used to find P(X > x) by subtracting the CDF from one. The expression for CDF is obtained by integrating the probability density function (pdf) from negative infinity to the value x.
Step-by-step explanation:
Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) for a continuous random variable is defined by P(X ≤ x), representing the probability that the random variable X takes on a value less than or equal to x. This function is fundamental in probability theory, because it describes the accumulation of probabilities up to a certain threshold. For a continuous distribution, the CDF gives us the area to the left of x under the probability density function (pdf) curve. Conversely, to find P(X > x) you would use the relationship P(X > x) = 1 - P(X ≤ x), which provides the area to the right of x.
To determine an expression for the CDF, you would integrate the pdf over the range from -∞ to x. The entire area under the probability density function curve is equal to one. For instance, if you have the pdf f(x), then the CDF F(x) is found by:
F(x) = ∫_{-∞}^{x} f(t) dt
This integral accumulates the probability from the far left of the distribution up to the value x, giving us the desired CDF.