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Problem 2: Let X be a continuous random variable with the followi probability density function S3a², for æ € [0, 1] 0 otherwise f(x) = • Compute E[X] and Var(X) • Compute P[0.25 < X < 0.75]

Problem 2: Let X be a continuous random variable with the followi probability density function S3a², for æ € [0, 1] 0 otherwise f(x) = • Compute E[X] and Var(X) • Compute P[0.25 < X < 0.75]

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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2. Problem 2: Let \( X \) be a continuous random variable with the following probability density function \[ f(x) = \begin{cases} 3x^2, & \text{for } x \in [0, 1] \\ 0, & \text{otherwise} \end{cases} \] - Compute \( E[X] \) and \( Var(X) \) - Compute \( P[0.25 \leq X \leq 0.75] \)
Transcribed Image Text:2. Problem 2: Let \( X \) be a continuous random variable with the following probability density function \[ f(x) = \begin{cases} 3x^2, & \text{for } x \in [0, 1] \\ 0, & \text{otherwise} \end{cases} \] - Compute \( E[X] \) and \( Var(X) \) - Compute \( P[0.25 \leq X \leq 0.75] \)
Expert Solution
Step 1

PDF of X is given by,

f(x) = 3x2for x∈[0,1]0otherwise

So, 

E(X)=∫-∞∞x f(x)dx=∫01x 3x2 dx=3∫01x3dx=3×x4401=3×14=0.75

Now, 

E(X2) = ∫01x2 3x2dx=3∫01x4dx=3×x5501=35 = 0.6

We know, 

Var(X)

= E(X2) - E(X)2 

=0.6 - 0.752 

=0.0375

=

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