PyESemana13Clase1.pdf - E1 Verify that P(−x < X < x = 2 � F(x − 1 provided the density function of X is an even function Observe that this holds in

# PyESemana13Clase1.pdf - E1 Verify that P(−x < X < x = 2 �...

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E1. Verify that P ( - x < X < x ) = 2 · F ( x ) - 1, provided the density function of X is an even function. Observe that this holds in particular for the standard normal distribution. P ( - x < X < x ) = P ( X < x ) - P ( X ≤ - x ) = F ( x ) - F ( - x ) = F ( x ) - [ 1 - F ( x ) ] = 2 · F ( x ) - 1
E2. A random variable X has Beta distribution with parameters a , b > 0, X d = B ( a , b ), provided f ( x ) = Γ( a + b ) Γ( a )Γ( b ) x a - 1 (1 - x ) b - 1 , 0 < x < 1 . Let X d = B (4 , 2) be the reservoir proportion sold per week in certain gas station. Find the probability of selling at least 90 % of the reservoir during a random week. P ( X 0 , 9) = Z 1 0 , 9 Γ(6) Γ(4)Γ(2) x 3 (1 - x ) dx = 5! 3! · 1! x 4 4 - x 5 5 1 0 , 9 0 , 08146
E3. It is known that certain bus will arrive at some time uniformly distributed between 0600 and 0630. Assuming at 0615 the bus has not yet arrived, what is the probability of waiting at least an additional 10 minutes? T : arrival time: T d = U (0 , 30) P ( T > 25 | T > 15) = P ( T > 25 , T > 15) P ( T > 15) = P ( T > 25) P ( T > 15) = Z 30 25 1 30 dt Z 30 15 1 30 dt = 5 15 = 1 3
E4. The Value at Risk (VAR) of an investment is the value v such that there is only a 1 % chance that the loss from the investment will exceed v . If the gain
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