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). 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). 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).

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). 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