A random variable X has a uniform distribution over (-3,3). Compute P(|X|<2)
Answers
Therefore the required probability of P( |X| < 2 ) is 2/3 or 0.67.
Given:
A random variable X has a uniform distribution over (-3,3).
To Find:
P(|X|<2) =?
Solution:
The given question can be easily solved using the below-shown approach.
Given that a random variable X has a uniform distribution over (-3,3).
where a = -3 and b = 3 and k = 1 / ( b-a ) = 1 / ( 3 - (-3 )) = 1/ ( 3 + 3 ) = 1 /6
In Uniform Distribution, the Probability function is given by,
⇒ f ( x ) = k = 1 / ( b-a ) if { a < x < b }; Otherwise f ( x ) = 0
∴ f ( x ) = k = 1/6
Now, |X| < 2 ⇒ ± X < 2 ⇒ X < 2 and - X < 2 ⇒ X < 2 and X > -2
∴ |X| < 2 = X < 2 and X > -2
⇒ P ( |X| < 2 ) = P ( X < 2 ) and P ( X > -2 ) = P ( 2 < X < -2 )
The Probability function for the random variable X which has a Uniform Distribution over ( -3, 3 ) is drawn and the image is attached. Kindly see the image for reference from here on.
From the image,
⇒ P ( 2 < X < -2 ) = Area of the hatched portion ( Rectangle )
⇒ P ( 2 < X < -2 ) = 4 × ( 1/6 ) = 2/3
Therefore the required probability of P( |X| < 2 ) is 2/3 or 0.67.
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