Math / Data & Statistics

Question6. The random variable $X$ has a distribution with density $g(x)=C x^{-3} 1_{[1, \infty)}(x)$ . a) Determine $C$ . b) Determine $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ .

Studdy Solution

STEP 1

1. The random variable $X$ has a probability density function (pdf) given by $g(x) = C x^{-3}$ for $x \geq 1$ .
2. The indicator function $1_{[1, \infty)}(x)$ ensures that the pdf is zero for $x < 1$ .
3. The constant $C$ must be determined such that the total probability integrates to 1 over the support of $X$ .

STEP 2

1. Determine the constant $C$ by integrating the pdf over its support.
2. Calculate $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ .

STEP 3

Set up the integral of the pdf over its support to equal 1:
$\int_{1}^{\infty} C x^{-3} \, dx = 1$

STEP 4

Integrate the function:
$C \int_{1}^{\infty} x^{-3} \, dx = C \left[ \frac{x^{-2}}{-2} \right]_{1}^{\infty} = C \left( 0 + \frac{1}{2} \right) = \frac{C}{2}$

STEP 5

Solve for $C$ :
$\frac{C}{2} = 1$
$C = 2$

STEP 6

Calculate $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ :
Since the support of $X$ is $[1, \infty)$ , we calculate:
$\mathbb{P}\left(X \in [1, 3]\right) = \int_{1}^{3} 2 x^{-3} \, dx$

STEP 7

Integrate the function:
$2 \int_{1}^{3} x^{-3} \, dx = 2 \left[ \frac{x^{-2}}{-2} \right]_{1}^{3} = 2 \left( \frac{-1}{2 \cdot 3^2} + \frac{1}{2} \right)$
$= 2 \left( \frac{-1}{18} + \frac{1}{2} \right)$
$= 2 \left( \frac{-1 + 9}{18} \right)$
$= 2 \left( \frac{8}{18} \right)$
$= 2 \left( \frac{4}{9} \right)$
$= \frac{8}{9}$
The constant $C$ is:
$\boxed{2}$
The probability $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ is:
$\boxed{\frac{8}{9}}$

Was this helpful?

Question6. The random variable $X$ has a distribution with density $g(x)=C x^{-3} 1_{[1, \infty)}(x)$ . a) Determine $C$ . b) Determine $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ .

Studdy Solution

STEP 1

STEP 2

1. Determine the constant $C$ by integrating the pdf over its support.
2. Calculate $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ .

STEP 3

Set up the integral of the pdf over its support to equal 1:
$\int_{1}^{\infty} C x^{-3} \, dx = 1$

STEP 4

Integrate the function:
$C \int_{1}^{\infty} x^{-3} \, dx = C \left[ \frac{x^{-2}}{-2} \right]_{1}^{\infty} = C \left( 0 + \frac{1}{2} \right) = \frac{C}{2}$

STEP 5

Solve for $C$ :
$\frac{C}{2} = 1$
$C = 2$

STEP 6

Calculate $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ :
Since the support of $X$ is $[1, \infty)$ , we calculate:
$\mathbb{P}\left(X \in [1, 3]\right) = \int_{1}^{3} 2 x^{-3} \, dx$

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question6. The random variable XXX has a distribution with density g(x)=Cx−31[1,∞)(x)g(x)=C x^{-3} 1_{[1, \infty)}(x)g(x)=Cx−31[1,∞)​(x). a) Determine CCC. b) Determine P(X∈[12,3])\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)P(X∈[21​,3]).

Studdy Solution

STEP 1

STEP 2

1. Determine the constant C C C by integrating the pdf over its support.2. Calculate P(X∈[12,3])\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)P(X∈[21​,3]).

STEP 3

Set up the integral of the pdf over its support to equal 1:∫1∞Cx−3 dx=1 \int_{1}^{\infty} C x^{-3} \, dx = 1 ∫1∞​Cx−3dx=1

STEP 4

Integrate the function:C∫1∞x−3 dx=C[x−2−2]1∞=C(0+12)=C2 C \int_{1}^{\infty} x^{-3} \, dx = C \left[ \frac{x^{-2}}{-2} \right]_{1}^{\infty} = C \left( 0 + \frac{1}{2} \right) = \frac{C}{2} C∫1∞​x−3dx=C[−2x−2​]1∞​=C(0+21​)=2C​

STEP 5

Solve for C C C:C2=1 \frac{C}{2} = 1 2C​=1C=2 C = 2 C=2

STEP 6

Calculate P(X∈[12,3])\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)P(X∈[21​,3]):Since the support of X X X is [1,∞) [1, \infty) [1,∞), we calculate:P(X∈[1,3])=∫132x−3 dx \mathbb{P}\left(X \in [1, 3]\right) = \int_{1}^{3} 2 x^{-3} \, dx P(X∈[1,3])=∫13​2x−3dx

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question6. The random variable $X$ has a distribution with density $g(x)=C x^{-3} 1_{[1, \infty)}(x)$ . a) Determine $C$ . b) Determine $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ .

1. Determine the constant $C$ by integrating the pdf over its support.
2. Calculate $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ .

Set up the integral of the pdf over its support to equal 1:
$\int_{1}^{\infty} C x^{-3} \, dx = 1$

Integrate the function:
$C \int_{1}^{\infty} x^{-3} \, dx = C \left[ \frac{x^{-2}}{-2} \right]_{1}^{\infty} = C \left( 0 + \frac{1}{2} \right) = \frac{C}{2}$

Solve for $C$ :
$\frac{C}{2} = 1$
$C = 2$

Calculate $\mathbb{P}\left(X \in\left[\frac{1}{2}, 3\right]\right)$ :
Since the support of $X$ is $[1, \infty)$ , we calculate:
$\mathbb{P}\left(X \in [1, 3]\right) = \int_{1}^{3} 2 x^{-3} \, dx$