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A Set of Open Resources for MATH 105 at UBC

- Home
- About
- Discrete Random Variables
- Continuous Random Variables
- 2.1 – The Cumulative Distribution Function
- 2.2 – A Simple Example
- 2.3 – The Probability Density Function
- 2.4 – A Simple PDF Example
- 2.5 – Some Common Continuous Distributions
- 2.6 – The Normal Distribution
- 2.7 – A Geometric Problem
- 2.8 – Expected Value, Variance, Standard Deviation
- 2.9 – Example
- 2.10 – Lesson 2 Summary

- Additional Problems

Previous: 2.3 – The Probability Density Function

Next: 2.5 – Some Common Continuous Distributions

Let *f*(*x*) = *k*(3*x*^{2} + 1).

- Find the value of
*k*that makes the given function a PDF on the interval 0 ≤*x*≤ 2. - Let
*X*be a continuous random variable whose PDF is*f*(*x*). Compute the probability that*X*is between 1 and 2. - Find the distribution function of
*X*. - Find the probability that
*X*is*exactly*equal to 1.

Therefore, *k* = 1/10.

Notice that *f*(*x*) ≥ 0 for all *x*. Also notice that we can rewrite this PDF in the obvious way so that it is defined for all real numbers:

Using our value for *k* from Part 1:

Therefore, Pr(1 ≤ *X* ≤ 2) is 4/5.

Using the Fundamental Theorem of Calculus, the CDF of *X* at *x* in [0,2] is

We can also easily verify that *F*(*x*) = 0 for all *x* < 0 and that *F*(*x*) = 1 for all *x* > 2.

Since *X* is a continuous random variable, we immediately know that the probability that it equals any one particular value must be zero. More directly, we compute

source: http://wiki.ubc.ca/Science:MATH105_Probability/Lesson_2_CRV/2.06_A_Simple_PDF_Example

### Navigation

- Home
- About
- Discrete Random Variables
- Continuous Random Variables
- 2.1 - The Cumulative Distribution Function
- 2.2 - A Simple Example
- 2.3 - The Probability Density Function
- 2.4 - A Simple PDF Example
- 2.5 - Some Common Continuous Distributions
- 2.6 - The Normal Distribution
- 2.7 - A Geometric Problem
- 2.8 - Expected Value, Variance, Standard Deviation
- 2.9 - Example
- 2.10 - Lesson 2 Summary

- Additional Problems

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### Acronyms

Throughout this website, the following acronyms are used.

PDF = probability distribution function

CDF = cumulative distribution function

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