Variables. a. If P(X > c) = α, determine P(X ≤ c) in terms of α. b. Suppose that P(Y > c) = α/2 and P(Y < −c) = P(Y > c). Determine P(−c ≤ Y ≤ c) in terms of α. c. Suppose that P(−c ≤ T ≤ c) = 1 − α and also suppose that P(T < −c) = P(T > c). Find P(T > c) in terms of α.

a) P(X ≤ c) = 1 - a

b) P(−c ≤ Y ≤ c) = 1 - 1.5a

c) P ( T > c ) = a/2

Explanation:

Given:-

- Let c > 0 and 0 < a < 1.

- Let X, Y & T be random variables.

Find:-

If P(X > c) = α, determine P(X ≤ c) in terms of α.

Solution:-

- To find the probability about the same constant "c" but with inverted signs " >" to "≤ " we will make use of total probability i.e 1 and subtract the given probability of any sign to get the probability with respect to other sign.

                 P(X ≤ c) = 1 - P(X > c)

                 P(X ≤ c) = 1 - a

Find:-

b. Suppose that P(Y > c) = α/2 and P(Y < −c) = P(Y > c).

Determine P(−c ≤ Y ≤ c) in terms of α.

Solution:-

- To find the probability between the same limit "c" we will break the relation into two parts P (Y < -c) and P (Y ≤ c)

                 P(−c ≤ Y ≤ c) = P ( Y ≤ c ) - P ( Y < -c )

                 P(−c ≤ Y ≤ c) = ( answer to part a ) - P(Y > c)

                 P(−c ≤ Y ≤ c) = (1 - a) - a/2

                 P(−c ≤ Y ≤ c) = 1 - 1.5a

Find:-

c. Suppose that P(−c ≤ T ≤ c) = 1 − α and also

suppose that P(T < −c) = P(T > c). Find P(T > c) in terms of α.

Solution:-

- To find the probability between the same limit "c" we will break the relation into two parts P (T < -c) and P (T ≤ c)

                 P(−c ≤ T≤ c) = P ( T ≤ c ) - P ( T < -c )

                 1 - a  = P ( T ≤ c ) - P(T > c)

                 1 - a  = [1 - P ( T > c )] - P(T > c)

                 - a  = - 2 P ( T > c )

                 P ( T > c ) = a/2