Question - Probability Calculation Using Probability Density Function
Solution:
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.
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