Suppose that X is a uniformly distributed random variable ov

Suppose that X is a uniformly distributed random variable over [-a,a] where a>0. Whenever possible, determine the value of a so that the following are satisfied:

1) P(X>1) = 1/3

2) P(X>1) = 1/2

3) P(X<1/2) = 0.7

Solution

f(x)=1/(2a) for a>0

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1) P(X>1) = 1/3

--> (a-1)/(2a) = 1/3

--> a-1 = (1/3)*(2a) = 2a/3

--> a-(2/3)a = 1

--> (1/3)a =1

SO a= 3

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2) P(X>1) = 1/2

--> (a-1)/(2a) = 1/2

--> a-1 = (1/2)*2a = a

So not soulation for a

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3) P(X<1/2) = 0.7

--> (1/2 -(-a))/(2a) = 0.7

--> (1/2 +a) / (2a) = 0.7

--> 1/2 +a = 0.7*(2a) = 1.4a

--> 1.4a - a= 0.5

--> 0.4a= 0.5

So a= 0.5/0.4 =1.25