Final answer:
The CDF of a uniformly distributed random variable X on the interval [2, 6] is 0 for x < 2, (x-2)/4 for 2 ≤ x ≤ 6, and 1 for x > 6. The expected value, EX, of this distribution is 4.
Step-by-step explanation:
When we choose a real number uniformly at random in the interval [2, 6] and call it X, we are defining a uniform distribution over the interval. The Cumulative Distribution Function (CDF), F_X(x), for a uniform distribution on the interval [a,b] is given by:
F_X(x) =
0 for x < a,
(x-a)/(b-a) for a ≤ x ≤ b,
1 for x > b.
In this case, a = 2 and b = 6. So for the given interval [2, 6]:
F_X(x) =
0 for x < 2,
(x-2)/4 for 2 ≤ x ≤ 6,
1 for x > 6.
To find the expected value EX of a uniform distribution on [a,b], we use the formula:
EX = (a + b) / 2
Therefore, for our interval [2, 6]:
EX = (2 + 6) / 2 = 4
The expected value of X is therefore 4.