152k views
4 votes
Determine an expression for the cumulative distribution function (CDF).

User Darksky
by
8.6k points

1 Answer

2 votes

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.

User Caolan
by
8.2k points

Related questions

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.