Problem

Use the probability density function given below to find the indicated probabilities. Illustrate each probability with a graph. \[ f(x)=\left\{\begin{array}{ll} \frac{1}{(1+x)^{2}} & \text { if } x \geq 0 \\ 0 & \text { otherwise } \end{array}\right. \] (A) $P(1 \leq X \leq 5)$ (B) $P(X>2)$ (C) $P(X \leq 5)$ (A) $P(1 \leq X \leq 5)=\square$ (Type an integer or a simplified fraction.)

Solution

Step 1 :The question is asking for the probability that a random variable X, which follows the given probability density function, falls within certain ranges. The probability that a continuous random variable falls within a certain range is given by the integral of the probability density function over that range.

Step 2 :For part (A), we need to find the probability that X is between 1 and 5. This is given by the integral from 1 to 5 of the function \(f(x)\).

Step 3 :The result of the integral is approximately 0.333, which represents the probability that the random variable X falls between 1 and 5.

Step 4 :Final Answer: \(P(1 \leq X \leq 5)=\boxed{\frac{1}{3}}\)

From Solvely APP
Source: https://solvelyapp.com/problems/8529/

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