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February 2nd, 2016, 12:45 PM   #1
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Random variable with continuous distribution

Hello,

I'm stuck with following problem:

Random variable X with the continuous distribution for some arbitrary constant a > 0:

$\displaystyle p(x) = 3/4a³(a² - x²)$ if |x| < a otherwise 0

i. Draw the distribution
ii. Show that p(x) is a valid distribution

Some hints how to get started with the problem would be appreciated!
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February 2nd, 2016, 02:49 PM   #2
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Can you clarify $p(x)$?
Is it $\dfrac{3}{4a^2(a^2 - x^2)}$, $\dfrac{3}{4a^2}(a^2 - x^2)$, $\dfrac{3}{4}a^2(a^2 - x^2)$?



A small latex tip; If you want to make your exponents more visible, use ^n.
For example, the code a^n will give $a^n$.
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February 2nd, 2016, 04:41 PM   #3
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Quote:
Originally Posted by Prob View Post
Hello,

I'm stuck with following problem:

Random variable X with the continuous distribution for some arbitrary constant a > 0:

$\displaystyle p(x) = 3/4a³(a² - x²)$ if |x| < a otherwise 0

i. Draw the distribution
ii. Show that p(x) is a valid distribution

Some hints how to get started with the problem would be appreciated!
For i), just substitute, say, a = 1 and plot some points. (Or use use the first and second derivative tests - your choice lol.)

For ii), you have to show that the area under the curve = 1, i.e. integrate p(x) from -a to a.
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February 2nd, 2016, 09:25 PM   #4
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p(x) is:

$\displaystyle p(x) = \frac{3}{4a^{3}}(a^2-x^2)$
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