Probability Calculation Using Probability Density Function
The image shows a graph with a density function \( f(x) \) for a random variable X and asks to find the probability that X is between 1 and 4, \( P(1 < X < 4) \).
The graph depicts a triangle that represents the probability density function, with the base of the triangle spanning from x = 0 to x = 4 and the height at f(x) = 1/2 at x = 0. The area under the curve of a probability density function \( f(x) \) between two points a and b gives the probability \( P(a < X < b) \).
To find \( P(1 < X < 4) \), we will calculate the area of the triangle between x = 1 and x = 4.
The original triangle's base is 4 units long (from x = 0 to x = 4), and the height is \( f(x) = 1/2 \). The area of the full triangle is:
\( \text{Area} = \frac{1}{2} (\text{base}) (\text{height}) = \frac{1}{2} (4) \left(\frac{1}{2}\right) = 1 \)
Now, we need only the area from x = 1 to x = 4. To find this portion, we subtract the area of the smaller triangle from x = 0 to x = 1 from the total area. The smaller triangle has a base of 1 unit and the same height of \( f(x) = 1/2 \). The area of this smaller triangle is:
\( \text{Area}_{\text{small}} = \frac{1}{2} (\text{base of small triangle}) (\text{height}) = \frac{1}{2} (1) \left(\frac{1}{2}\right) = \frac{1}{4} \)
Now subtract the area of the small triangle from the total area to find the desired probability:
\( P(1 < X < 4) = \text{Area}_{\text{total}} - \text{Area}_{\text{small}} = 1 - \frac{1}{4} = \frac{3}{4} \)
So, the probability that X is between 1 and 4 is \( \frac{3}{4} \) or 0.75.
