Let X be a continuous random variable and $f(x)$ is defined as : $$f(x)=\begin{cases}kx(1-x)^{10}, & 0<x<1\\0, & \text{otherwise}\end{cases}$$ find the value of k.

> [!TLDR] > Uses the normalisation condition $\int_{-\infty}^{\infty}f(x)\,dx=1$ together with the Beta-function integral to solve for $k$. > > 1. Since $f$ is a probability density function, the total area under it must equal $1$: $\displaystyle\int_0^1 kx(1-x)^{10}\,dx=1$ (as $f(x)=0$ outside $(0,1)$). > 2. Pull the constant out: $k\displaystyle\int_0^1 x(1-x)^{10}\,dx=1$. > 3. Evaluate $\displaystyle\int_0^1 x(1-x)^{10}\,dx$ using the substitution $u=1-x$ (so $x=1-u$, $dx=-du$; limits flip from $x:0\to1$ to $u:1\to0$): $\displaystyle\int_0^1(1-u)u^{10}\,du=\int_0^1\left(u^{10}-u^{11}\right)du$. …

Let X be a continuous random variable and f(x) is defined as : f(x)= kx(1-x)¹⁰, 0<x<1; 0, otherwise find the v

Question 84 of 105

f(x)

Puducherry TnboardTamil Nadu HSC (DGE) Board 2020Subjective· 3mImportance★★★★★

80% · 84/105 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Uses the normalisation condition $\int_{-\infty}^{\infty}f(x)\,dx=1$ together with the Beta-function integral to solve for $k$ .

Since $f$ is a probability density function, the total area under it must equal $1$ : $\displaystyle\int_0^1 kx(1-x)^{10}\,dx=1$ (as $f(x)=0$ outside $(0,1)$ ).
Pull the constant out: $k\displaystyle\int_0^1 x(1-x)^{10}\,dx=1$ .
Evaluate $\displaystyle\int_0^1 x(1-x)^{10}\,dx$ using the substitution $u=1-x$ (so $x=1-u$ , $dx=-du$ ; limits flip from $x:0\to1$ to $u:1\to0$ ): $\displaystyle\int_0^1(1-u)u^{10}\,du=\int_0^1\left(u^{10}-u^{11}\right)du$ . …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ28 — A random variable X has the following probability mass function : | X …→Next questionQ30 — If f(x)= 2x, 0≤ x≤ a; 0, otherwise is a probability density function o…

Q.Let X be a continuous random variable and $f(x)$ is defined as :
[!FORMULA] $f(x)=\begin{cases}kx(1-x)^{10}, & 0<x<1\\0, & \text{otherwise}\end{cases}$
find the value of k.

Concept understanding — Probability Density Function & Cumulative Distribution

Q.Let X be a continuous random variable and f(x)f(x)f(x) is defined as : [!FORMULA] f(x)={kx(1−x)10,0<x<10,otherwisef(x)=\begin{cases}kx(1-x)^{10}, & 0<x<1\\0, & \text{otherwise}\end{cases}f(x)={kx(1−x)10,0,​0<x<1otherwise​ find the value of k.

🎓Unlock everything free for 14 days

Q.Let X be a continuous random variable and $f(x)$ is defined as :
[!FORMULA] $f(x)=\begin{cases}kx(1-x)^{10}, & 0<x<1\\0, & \text{otherwise}\end{cases}$
find the value of k.

Unlock everything free for 14 days