Finding cdf and pdf of variable.

In summary, the conversation discusses finding the cdf and pdf of the minimum variable Y from four independent random variables X1, X2, X3, and X4 with a given pdf. The attempt at a solution involves using the formula P(Y>or=y) and integrating over the pdf to find the correct answer.
  • #1
cookiesyum
78
0

Homework Statement



Let X1 X2 X3 and X4 be four independent random variables, each with pdf f(x) = 3(1-x)2, 0<x<1, zero elsewhere. If Y is the minimum of these four variables, find the cdf and pdf of Y.

The Attempt at a Solution



P(Y<or= y)

= 1 - P(Y>y)
= 1 - P(X1>y, X2>y, X3>y, X4>y)
= 1 - P(X1>y)P(X2>y)P(X3>y)P(X4>y)
= 1 - [3(1-x)2]4

which does NOT equal (1-y)12, which is the answer in the back of the book. Don't know where I'm going wrong...
 
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  • #2
What is P(x_1 > y)? It's certainly not that. Maybe you should try integrating over f instead.
 
  • #3
clamtrox said:
What is P(x_1 > y)? It's certainly not that. Maybe you should try integrating over f instead.

Got it! Thanks so much for the help.
 

1. What is the difference between cdf and pdf?

CDF stands for cumulative distribution function, and it gives the probability that a random variable takes on a value less than or equal to a given value. PDF stands for probability density function, and it gives the probability of a random variable taking on a specific value. In other words, CDF shows the probability of being less than or equal to a value, while PDF shows the probability of being exactly at that value.

2. How do you find the cdf of a continuous variable?

To find the cdf of a continuous variable, you need to integrate the pdf of the variable from negative infinity to the desired value. This integral will give you the area under the curve up to that value, which represents the probability of the variable being less than or equal to that value.

3. Can you find the pdf from the cdf?

Yes, you can find the pdf from the cdf by taking the derivative of the cdf function. This will give you the slope of the cdf curve at a specific point, which is equivalent to the pdf at that point.

4. What is the range of values for cdf and pdf?

The range of values for cdf is from 0 to 1, as it represents the cumulative probability of a random variable being less than or equal to a certain value. The range of values for pdf is from 0 to infinity, as it represents the probability of a random variable taking on a specific value.

5. Why is it important to find the cdf and pdf of a variable?

Finding the cdf and pdf of a variable is important because it allows us to understand the behavior and characteristics of the variable. It also helps us in making predictions and calculating probabilities related to the variable. Additionally, cdf and pdf are essential in many statistical analyses and modeling techniques.

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