A random variable has a triangular probability density function with a = 50, b = 375, and m = 250.

What is the probability that the random variable will assume a value greater than 330? If required, round your answer to four decimal places.


A triangular probability density function (PDF) is defined by three parameters:

- \( a \): the minimum value (50 in this case)
- \( b \): the maximum value (375 in this case)
- \( m \): the mode (250 in this case)

The PDF of a triangular distribution can be broken down into two parts: one for the interval \( [a, m] \) and the other for the interval \( [m, b] \). In this case, since we are interested in the probability that the random variable \( X \) assumes a value greater than 330, we need to calculate \( P(X > 330) \).

First, the formula for the triangular PDF is:

\[
f(x) = 
\begin{cases} 
\frac{2(x - a)}{(b - a)(m - a)} & \text{for } a \leq x \leq m \\
\frac{2(b - x)}{(b - a)(b - m)} & \text{for } m \leq x \leq b 
\end{cases}
\]

To calculate \( P(X > 330) \), we use the cumulative distribution function (CDF), denoted as \( F(x) \), and the complementary probability:

\[
P(X > 330) = 1 - F(330)
\]

Let's calculate this step by step.

For \( x \) in the interval \( [m, b] \), the CDF \( F(x) \) is given by:

\[
F(x) = 1 - \frac{(b - x)^2}{(b - a)(b - m)}
\]

Plugging in the values for \( x = 330 \), \( a = 50 \), \( b = 375 \), and \( m = 250 \):

\[
F(330) = 1 - \frac{(375 - 330)^2}{(375 - 50)(375 - 250)}
\]

Now let's compute this.The probability that the random variable will assume a value greater than 330 is approximately \( 0.0498 \), or 4.98%.

Would you like more details on this solution, or do you have any questions?

Here are 5 related questions for deeper understanding:
1. How is the triangular distribution used in real-world applications?
2. What is the interpretation of the mode \( m \) in a triangular distribution?
3. How does changing the value of \( m \) affect the shape of the triangular distribution?
4. Can you derive the formula for the CDF for both intervals \( [a, m] \) and \( [m, b] \)?
5. How would you calculate the probability for a value less than 330?

**Tip**: For any distribution, the CDF represents the probability that the variable will take a value less than or equal to a given point.

A random variable has a triangular probability density function with a = 50, b = 375, and m = 250. What is the probability that the random variable will assume a value greater than 330? If required, round your answer to four decimal places.

Explore how to calculate the probability that a random variable with a triangular probability density function (a = 50, b = 375, m = 250) will assume a value greater than 330. Learn about the triangular distribution, its PDF, CDF, and the steps to determine P(X > 330). Suitable for college-level probability students.

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