Final answer:
In a continuous probability distribution, the probability of the variable taking a specific value is zero. To find the probability for a range of values in an exponential distribution, the cumulative distribution function is utilized. For the exponential distribution, P(1 < X < 4) can be found using the CDF, but P(X = 3) is simply 0.
Step-by-step explanation:
The question provided seems to pertain to a continuous probability distribution, specifically an exponential distribution. For a continuous distribution, the probability at a single point, P(X = x), is always zero, because the area under the curve at a point is zero. For an exponential distribution, if f(x) = e-x for x > 0, then the cumulative distribution function is given by P(X < x) = 1 - e-mx, where m is the rate parameter of the exponential distribution. To find P(X > x), you would use the complement of the cumulative distribution function, which is P(X > x) = e-mx.
To calculate P(1 < x < 4) for an exponential distribution, you would find P(X < 4) and P(X < 1) and subtract the latter from the former. That is P(1 < x < 4) = P(X < 4) - P(X < 1). As noted in the provided question, the cumulative distribution for an exponential distribution is P(X < x) = 1 - e-0.5x, so you would substitute x with 1 and 4 respectively, and calculate the difference.
Since the probability density at a specific point for a continuous distribution is 0, we can immediately state that P(X = 3) = 0.