Question

The Cumulative distribution function of random variable X is:

F_X(x) =

0, x < -1
(x + 1) / 4, -1 ≤ x < 1
1, x ≥ 1
(a) Find P[X ≤ 1].
(b) Find P[X < 1].
(c) Find P[X = 1].
(d) Determine the probability density function, f_X(x).

Which of the following correctly answers the above questions?

A) P[X ≤ 1] = 0.75, P[X < 1] = 0.25, P[X = 1] = 0, and f_X(x) = 0
B) P[X ≤ 1] = 0.5, P[X < 1] = 0.5, P[X = 1] = 0, and f_X(x) = 0
C) P[X ≤ 1] = 0.25, P[X < 1] = 0.75, P[X = 1] = 0, and f_X(x) = 1/4
D) P[X ≤ 1] = 1, P[X < 1] = 0.25, P[X = 1] = 0, and f_X(x) = (1/4) * δ(x - 1)

Answer 1

The cumulative distribution function (CDF) of a random variable X is used to find probabilities of different intervals. (a) P[X ≤ 1] = 0.5. (b) P[X < 1] = 0.5. (c) P[X = 1] = 0. (d) The probability density function (pdf) is f_X(x) = 1/4. The correct answer is D.

Step-by-step explanation:

The cumulative distribution function (CDF) of a random variable X is defined as the probability that X is less than or equal to a given value x. We can use the CDF to find probabilities for different intervals.

(a) To find P[X ≤ 1], we use the CDF. Since -1 ≤ 1, we use the function (x + 1) / 4 for the interval -1 ≤ x < 1. Plugging in x = 1, we get P[X ≤ 1] = (1 + 1) / 4 = 2/4 = 1/2 = 0.5.

(b) To find P[X < 1], we subtract the probability of the exact value x = 1 from P[X ≤ 1]. Since P[X = 1] = 0, P[X < 1] = (1 + 1) / 4 - 0 = 2/4 = 1/2 = 0.5.

(c) P[X = 1] = 0, as given in the cumulative distribution function.

(d) To find the probability density function (pdf), we take the derivative of the CDF with respect to x. For -1 ≤ x < 1, the CDF is (x + 1) / 4. Taking the derivative, we get f_X(x) = 1/4.

Therefore, the correct answer is D) P[X ≤ 1] = 1, P[X < 1] = 0.5, P[X = 1] = 0, and f_X(x) = (1/4) * δ(x - 1).