For a continuous random variable, the probability that X equals a specific value is?

For a continuous random variable, the probability that X equals a specific value is?

• A. It is always zero
• B. It can be positive
• C. It depends on c
• D. It is always 1

Answer: (A)

For a continuous random variable, probabilities are assigned to intervals, not to exact points, because the area under the density at a single point has no width. The probability that X equals a specific value x is the integral of the density over that single point, which has zero measure, so P(X = x) = 0 for any continuous distribution (as long as there are no discrete point masses). This is why the probability is always zero, regardless of parameter values like c. The other possibilities would require a point mass (positive probability at a single value), a dependence on a parameter that changes that mass, or certainty, none of which applies to a purely continuous variable.

For a continuous random variable, probabilities are assigned to intervals, not to exact points, because the area under the density at a single point has no width. The probability that X equals a specific value x is the integral of the density over that single point, which has zero measure, so P(X = x) = 0 for any continuous distribution (as long as there are no discrete point masses).

This is why the probability is always zero, regardless of parameter values like c. The other possibilities would require a point mass (positive probability at a single value), a dependence on a parameter that changes that mass, or certainty, none of which applies to a purely continuous variable.