In a discrete distribution P(x<4) is equivalent to what:

Question

a) P(x≤3)

(b) P(x≤5)

(c) P(0)+P(1)+P(2)+P(3)+P(4)

(d) 1-P(x>4)

Answer 1

If P(X = 4) > 0, We can easily rule out b, c), and d) For example, if we look at a subset of discrete distributions, P(X > 0) on non-negative integers, P(X < 4) = P(0) + P(1) + P(2) + P(3)

b) P(X <= 5) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) is not equal to P(0) + P(1) + P(2) + P(3) (the difference is P(4) + P(5))

c) P(0) + P(1) + P(2) + P(3) + P(4) not equal to P(X < 4) = P(0) + P(1) + P(2) + P(3), as the difference is P(4)

d) 1 - P(X > 4) = P(X <= 4) = P(0) + P(1) + P(2) + P(3) + P(4) (same as c)); the difference is again P(4)

The author may have intended a) to be the correct answer. If the discrete distribution is only defined on non-negative integers (e.g., , the Poisson, geometric, binomial, negative binomial, hyper-geometric), a) is true, as P(X <= 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(x = 3), and P(X <= 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(x = 3), the same.

Let's give a counter-example for a.

Let X be the random variable for the average roll of 2 fair dice. X takes on values 1, 3/2, 2, 5/2, 3, 7/2, 4. 9/2, 5, 11/2, and 6.

This is a discrete distribution.

P(X = 7/2) = 1/6. Thus, P(X < 4) = P(X <= 3) + P(X = 7/2) = P(X <= 3) + 1/6, so P(X = 3) is not equal to P(X < 4)

I'd love to know what book this problem comes from.