The density function of X is given by the rule f(x)= x for 0<x ≤1 2-x for 1<x<2

step 1 Hello, welcome to this lesson. In this lesson we have x that has a density function of f of x eq

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

The density function of $X$ is given by the rule $$ f(x)= \begin{cases}x & \text { for } 0<x \leq 1 \\ 2-x & \text { for } 1<x<2 \\ 0 & \text { elsewhere }\end{cases} $$ (a) find the CDF (b) prove that $f(x)$ satisfies the two requirements for a density function (c) find $P\left(\frac{1}{2}<\mathrm{X}<\frac{3}{2}\right)$.

The density function of $X$ is given by the rule
$$
f(x)= \begin{cases}x & \text { for } 0<x \leq 1 \\ 2-x & \text { for } 1<x<2 \\ 0 & \text { elsewhere }\end{cases}
$$
(a) find the CDF
(b) prove that $f(x)$ satisfies the two requirements for a density function
(c) find $P\left(\frac{1}{2}<\mathrm{X}<\frac{3}{2}\right)$.

Key Concepts

Probability Calculation for Intervals

Calculating the probability that a continuous random variable falls within a specific interval can be done by integrating the probability density function over that interval or by taking the difference between the cumulative distribution function values at the endpoints of the interval. This method effectively captures the likelihood of the variable residing in that range.

Verification of Density Function Conditions

Verifying that a function is a valid probability density function involves checking that the function is non-negative for all values and that its total integral over the entire space equals one. These two conditions are essential to ensure that the function properly assigns probabilities to events within the distribution's domain.

Probability Density Function

A probability density function (pdf) describes how a continuous random variable's probabilities are distributed across possible values. It must be non-negative over its entire domain and integrate to one over that domain, ensuring that it represents a valid probability distribution.

Cumulative Distribution Function

The cumulative distribution function (CDF) of a random variable gives the probability that the variable takes on a value less than or equal to a given number. It is obtained by integrating the probability density function from the minimum possible value up to the point of interest, resulting in a non-decreasing function that ranges from 0 to 1.

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