I have very difficulty to understand and do this problem. Please explain in detail. Thanks. A probability density function on R is a function f : R → R satisfying (i) f(x) > 0 for all x E R and (ii) / f(x) dx 1. For which value(s) of k eR is the function f (x) = e¬a* Vk5 a probability density function? Explain.
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
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![**Understanding Probability Density Functions**
A probability density function (PDF) on the real numbers \(\mathbb{R}\) is a function \(f : \mathbb{R} \to \mathbb{R}\) that satisfies the following conditions:
(i) \(f(x) \geq 0\) for all \(x \in \mathbb{R}\).
(ii) \(\int_{-\infty}^{\infty} f(x) \, dx = 1\).
**Problem:**
Determine the value(s) of \(k \in \mathbb{R}\) for which the function
\[ f(x) = e^{-x^2} \sqrt[3]{\frac{1}{k^5}} \]
is a probability density function.
**Explanation:**
To ensure \(f(x)\) is a valid PDF, ensure that:
1. \(f(x)\) is non-negative for all \(x\).
2. The integral over the entire real line equals 1:
\[
\int_{-\infty}^{\infty} f(x) \, dx = \int_{-\infty}^{\infty} e^{-x^2} \sqrt[3]{\frac{1}{k^5}} \, dx = 1.
\]
You need to solve this equation to find the appropriate \(k\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcaa0f822-544e-4e71-bb85-904850361648%2F53c19610-0a13-4665-bc49-99ddc1f11f17%2Fsnlz0d_processed.png&w=3840&q=75)
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