Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![The given integral to evaluate is:
\[
\int_0^1 u^2 \left( \sqrt{u} + \sqrt[3]{u} \right) \, du
\]
Here, \( u \) is the variable of integration, and the limits of the integral are from \( 0 \) to \( 1 \). The integrand function is \( u^2 \left( \sqrt{u} + \sqrt[3]{u} \right) \), which involves both square and cube roots of \( u \) multiplied by \( u^2 \).
To solve the integral, follow these steps:
1. Distribute \( u^2 \) inside the parenthesis:
\[ u^2 (\sqrt{u} + \sqrt[3]{u}) = u^2 \sqrt{u} + u^2 \sqrt[3]{u} \]
2. Simplify \( u^2 \sqrt{u} \) and \( u^2 \sqrt[3]{u} \):
\[ u^2 \sqrt{u} = u^2 \cdot u^{1/2} = u^{2+1/2} = u^{5/2} \]
\[ u^2 \sqrt[3]{u} = u^2 \cdot u^{1/3} = u^{2+1/3} = u^{7/3} \]
3. Rewrite the integral as:
\[ \int_0^1 (u^{5/2} + u^{7/3}) \, du \]
4. Integrate each term separately:
\[ \int_0^1 u^{5/2} \, du + \int_0^1 u^{7/3} \, du \]
\[
\int u^{5/2} \, du = \frac{u^{5/2+1}}{5/2+1} = \frac{u^{7/2}}{7/2} = \frac{2}{7} u^{7/2} \Big|_0^1 = \frac{2}{7} \left(1^{7/2} - 0^{7/2} \right) = \frac{2}{7}
\]
\[
\int u^{7/3} \, du = \frac{u^{7/](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F139cb5f2-923a-4627-93ad-b9a40e985e19%2F0bec6d99-679c-4333-adb4-3cba85899188%2F1nqbmws.png&w=3840&q=75)
Step by step
Solved in 4 steps with 4 images
