Suppose that random variable
X
X is uniformly distributed between 0 and 10. Then find
P
(
X
+
10
X
≥
7
)
P(X+
X
10
​
≥7). (Write answer upto two decimal places)

Given that the random variable \( X \) is uniformly distributed between 0 and 10, we need to find the probability \( P\left(X + \frac{10}{X} \geq 7\right) \).

### Step 1: Write down the given information.
- \( X \sim U(0, 10) \) (Uniformly distributed between 0 and 10)
- We need to find \( P\left(X + \frac{10}{X} \geq 7\right) \).

### Step 2: Set up the inequality.
Let \( Y = X + \frac{10}{X} \).

We want to find the probability that \( Y \geq 7 \).

This inequality can be rearranged as:
\[
X + \frac{10}{X} \geq 7
\]

### Step 3: Solve the inequality.
First, multiply both sides by \( X \) (assuming \( X > 0 \)):
\[
X^2 + 10 \geq 7X
\]

Rearrange the inequality:
\[
X^2 - 7X + 10 \geq 0
\]

### Step 4: Solve the quadratic equation.
To solve \( X^2 - 7X + 10 = 0 \), we use the quadratic formula:
\[
X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -7 \), and \( c = 10 \).

Substituting the values:
\[
X = \frac{7 \pm \sqrt{(-7)^2 - 4(1)(10)}}{2(1)}
\]
\[
X = \frac{7 \pm \sqrt{49 - 40}}{2}
\]
\[
X = \frac{7 \pm \sqrt{9}}{2}
\]
\[
X = \frac{7 \pm 3}{2}
\]
So, the roots are:
\[
X = \frac{10}{2} = 5 \quad \text{and} \quad X = \frac{4}{2} = 2
\]

### Step 5: Determine the intervals of interest.
The quadratic expression \( X^2 - 7X + 10 \) is positive when \( X \) is in the intervals \( (-\infty, 2] \) or \( [5, \infty) \).

Since \( X \) is uniformly distributed between 0 and 10, the valid intervals for \( X \) are \( [0, 2] \) and \( [5, 10] \).

### Step 6: Calculate the probability.
The total range of \( X \) is from 0 to 10, so the length of the interval is 10.

The intervals where the inequality holds are \( [0, 2] \) and \( [5, 10] \). 

The combined length of these intervals is:
\[
2 - 0 + 10 - 5 = 7
\]

The probability is:
\[
P\left(X + \frac{10}{X} \geq 7\right) = \frac{7}{10} = 0.70
\]

Thus, the final answer is \( \boxed{0.70} \).

Would you like to explore more details or ask any questions? Here are five related questions you might find interesting:

1. How would the probability change if \( X \) were uniformly distributed between 0 and 20?
2. What is the expected value of \( X + \frac{10}{X} \) given that \( X \) is uniformly distributed between 0 and 10?
3. How would the solution change if the uniform distribution was on a different interval, say [2, 12]?
4. What are the implications of having \( X \) uniformly distributed, and how does it affect the probability distribution of \( Y = X + \frac{10}{X} \)?
5. How would you calculate \( P\left(X + \frac{20}{X} \geq 7\right) \) if \( X \) is still uniformly distributed between 0 and 10?

**Tip:** Understanding how to solve quadratic inequalities and interpreting them in the context of probability is crucial for solving problems involving non-linear inequalities with random variables.

Explore how to find the probability of a quadratic inequality involving a uniformly distributed random variable. Learn step-by-step with detailed calculations and explanations. Suitable for high school students studying probability and distributions.

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