https://file.aiquickdraw.com/mathbot/file/17260355208333nycibsz.jpg

The image shows a probability density function (PDF) \( f(x) \) of a random variable \( X \). The function is defined as:

\[
f(x) = 
\begin{cases}
kx, & 0 \leq x \leq 5 \\
k(x-3), & 5 \leq x \leq 10 \\
0, & \text{otherwise}
\end{cases}
\]

The questions associated with this function are:

1. (a) Find the value of \( k \).
2. (b) Find the expected value \( E(X) \).
3. (c) Find the variance \( \text{Var}(X) \).
4. (d) Find \( P(4 \leq X \leq 6) \).
5. (e) Sketch the graph of \( f(x) \).

Let's solve each part step-by-step:

### (a) Find the value of \( k \)

To find \( k \), we use the fact that the total area under the PDF must be 1:

\[
\int_{-\infty}^{\infty} f(x) \, dx = 1
\]

Since the function is only nonzero from \( 0 \) to \( 10 \), we split the integral into two parts:

\[
\int_0^5 kx \, dx + \int_5^{10} k(x - 3) \, dx = 1
\]

Calculating these integrals:

1. For the first part:

   \[
   \int_0^5 kx \, dx = k \left[\frac{x^2}{2}\right]_0^5 = k \cdot \frac{25}{2} = \frac{25k}{2}
   \]

2. For the second part:

   \[
   \int_5^{10} k(x - 3) \, dx = k \left[\frac{(x - 3)^2}{2}\right]_5^{10} = k \cdot \frac{49}{2} = \frac{49k}{2}
   \]

Adding both integrals:

\[
\frac{25k}{2} + \frac{49k}{2} = \frac{74k}{2} = 37k
\]

Setting this equal to 1:

\[
37k = 1 \implies k = \frac{1}{37}
\]

### (b) Find the expected value \( E(X) \)

The expected value \( E(X) \) is given by:

\[
E(X) = \int_{-\infty}^{\infty} x f(x) \, dx = \int_0^5 x(kx) \, dx + \int_5^{10} x k(x - 3) \, dx
\]

Substituting \( k = \frac{1}{37} \) and evaluating:

1. For the first part:

   \[
   \int_0^5 x \cdot \frac{x}{37} \, dx = \frac{1}{37} \int_0^5 x^2 \, dx = \frac{1}{37} \cdot \frac{x^3}{3} \Bigg|_0^5 = \frac{125}{111}
   \]

2. For the second part:

   \[
   \int_5^{10} x \cdot k(x - 3) \, dx = \frac{1}{37}
   \]




Given a random variable X with the probability density function f(x) defined as:
 f(x) = {kx, 0 ≤ x ≤ 5; k(x-3), 5 ≤ x ≤ 10; 0, otherwise}, find the following: 
a) The value of k.
b) The expected value E(X).
c) The variance Var(X).
d) P(4 ≤ X ≤ 6).
e) Sketch the graph.

Learn how to find the value of k, expected value, variance, and the probability in a given range for a random variable X defined by a piecewise probability density function. Suitable for undergraduate students learning probability theory.

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