For A Uniform Random Variable $U \sim U(7,12$\], Find The Probability $P(10\ \textless \ U\ \textless \ 11$\]:$P(10\ \textless \ U\ \textless \ 11) = \square$ (Round The Answer To 4 Decimal Places)

Introduction

In probability theory, a uniform random variable is a continuous random variable that has a constant probability density function (PDF) over a given interval. In this article, we will discuss how to find the probability of a uniform random variable $U$ falling within a specific range. We will use the example of a uniform random variable $U \sim U(7,12)$ to calculate the probability $P(10\ \textless \ U\ \textless \ 11)$.

Understanding Uniform Random Variables

A uniform random variable $U$ is said to be uniformly distributed over the interval $[a,b]$ if its probability density function (PDF) is given by:

$$f(x) = \begin{cases} \frac{1}{b-a} & \text{if } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}$$

In this case, the uniform random variable $U$ is distributed over the interval $[7,12]$. The probability density function of $U$ is given by:

$$f(x) = \begin{cases} \frac{1}{12-7} = \frac{1}{5} & \text{if } 7 \leq x \leq 12 \\ 0 & \text{otherwise} \end{cases}$$

Calculating the Probability

To calculate the probability $P(10\ \textless \ U\ \textless \ 11)$, we need to find the area under the probability density function of $U$ between the points $x=10$ and $x=11$. Since the probability density function is constant over the interval $[7,12]$, the area under the curve between $x=10$ and $x=11$ is simply the product of the probability density function and the length of the interval:

$$P(10\ \textless \ U\ \textless \ 11) = f(x) \cdot (11-10) = \frac{1}{5} \cdot 1 = \frac{1}{5}$$

However, this is not the final answer. We need to round the answer to 4 decimal places.

Rounding the Answer

To round the answer to 4 decimal places, we can use the following formula:

$$P(10\ \textless \ U\ \textless \ 11) = \frac{1}{5} = 0.2$$

Conclusion

In this article, we discussed how to find the probability of a uniform random variable $U$ falling within a specific range. We used the example of a uniform random variable $U \sim U(7,12)$ to calculate the probability $P(10\ \textless \ U\ \textless \ 11)$. We found that the probability is equal to $\boxed{0.2}$.

Related Topics

• Uniform Distribution: A uniform distribution is a continuous probability distribution where every possible outcome has an equal probability of occurring.
• Probability Density Function: A probability density function (PDF) is a function that describes the probability distribution of a continuous random variable.
• Random Variables: A random variable is a variable whose value is determined by chance or probability.

References

• Wikipedia: Uniform Distribution (Continuous)
• Khan Academy: Probability Density Function (PDF)
• Math Is Fun: Random Variables
  Uniform Random Variable and Probability Calculation: Q&A =====================================================

Introduction

In our previous article, we discussed how to find the probability of a uniform random variable $U$ falling within a specific range. We used the example of a uniform random variable $U \sim U(7,12)$ to calculate the probability $P(10\ \textless \ U\ \textless \ 11)$. In this article, we will answer some frequently asked questions related to uniform random variables and probability calculation.

Q&A

Q: What is a uniform random variable?

A: A uniform random variable is a continuous random variable that has a constant probability density function (PDF) over a given interval.

Q: How do I calculate the probability of a uniform random variable falling within a specific range?

A: To calculate the probability of a uniform random variable falling within a specific range, you need to find the area under the probability density function of the variable between the points that define the range.

Q: What is the probability density function (PDF) of a uniform random variable?

A: The probability density function (PDF) of a uniform random variable is given by:

$$f(x) = \begin{cases} \frac{1}{b-a} & \text{if } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}$$

Q: How do I find the area under the probability density function of a uniform random variable?

A: To find the area under the probability density function of a uniform random variable, you need to multiply the probability density function by the length of the interval between the points that define the range.

Q: Can I use the same formula to calculate the probability of a uniform random variable falling within a specific range for any interval?

A: Yes, you can use the same formula to calculate the probability of a uniform random variable falling within a specific range for any interval. The formula is:

$$P(a\ \textless \ X\ \textless \ b) = \frac{1}{b-a} \cdot (b-a)$$

Q: What is the probability of a uniform random variable $U \sim U(7,12)$ falling within the range $[10,11]$?

A: The probability of a uniform random variable $U \sim U(7,12)$ falling within the range $[10,11]$ is:

$$P(10\ \textless \ U\ \textless \ 11) = \frac{1}{5} \cdot 1 = \frac{1}{5} = 0.2$$

Q: Can I use a calculator to calculate the probability of a uniform random variable falling within a specific range?

A: Yes, you can use a calculator to calculate the probability of a uniform random variable falling within a specific range. You can use the formula:

$$P(a\ \textless \ X\ \textless \ b) = \frac{1}{b-a} \cdot (b-a)$$

to calculate the probability.

Conclusion

In this article, we answered some frequently asked questions related to uniform random variables and probability calculation. We hope that this article has been helpful in understanding the concept of uniform random variables and how to calculate the probability of a uniform random variable falling within a specific range.

Related Topics

• Uniform Distribution: A uniform distribution is a continuous probability distribution where every possible outcome has an equal probability of occurring.
• Probability Density Function: A probability density function (PDF) is a function that describes the probability distribution of a continuous random variable.
• Random Variables: A random variable is a variable whose value is determined by chance or probability.

References

• Wikipedia: Uniform Distribution (Continuous)
• Khan Academy: Probability Density Function (PDF)
• Math Is Fun: Random Variables