The density function of X is given by f(x) =(2(3-x))/(9) for 0 ≤x ≤3, =0 e

step 1 On this problem, we are told that we are given a probability density function of one -half where

The density function of X is given by f(x) ...

The density function of $X$ is given by $$ \begin{aligned} f(x) & =\frac{2(3-x)}{9} & \text { for } 0 \leq x \leq 3, \\ & =0 & \text { elsewhere. } \end{aligned} $$ Without using integration, show that the area under the curve is equal to one and find $P(1<X<1.5)$ and $P(X>2)$. [Hint: Use the fact that the area within a triangle is equal to $\frac{1}{2}$ of the product of the base and the height.]

The density function of $X$ is given by
$$
\begin{aligned}
f(x) & =\frac{2(3-x)}{9} & \text { for } 0 \leq x \leq 3, \\
& =0 & \text { elsewhere. }
\end{aligned}
$$
Without using integration, show that the area under the curve is equal to one and find $P(1<X<1.5)$ and $P(X>2)$. [Hint: Use the fact that the area within a triangle is equal to $\frac{1}{2}$ of the product of the base and the height.]

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Probability Density Function (PDF)

A PDF is a function that describes the relative likelihood for a continuous random variable to occur at each point in its range. The area under the PDF over a given interval represents the probability that the random variable falls within that interval, and the total area across its support must equal one.

Normalization

Normalization is the property that the area under the PDF over its entire range is equal to one. This ensures that the probabilities assigned by the function are valid, as it guarantees that some outcome within the sample space will occur.

Geometric Probability

Geometric probability involves computing probabilities using geometric measures such as areas and lengths. When a PDF has a shape that conforms to simple geometric figures (like a triangle), the probability for an interval can be found by calculating the area of the corresponding geometric region.

Triangle Area Formula

The triangle area formula, which states that the area is one-half the product of its base and height, can be applied to find the area under a triangular section of a PDF. This approach provides a way to verify normalization and calculate probabilities without performing integration.

Transcript

Please give Ace some feedback