Chapter 5: Q.5.1 (page 212)

Let $X$ be a random variable with probability density function
$f (x) = \{\begin{cases} c (1 - x^{2}) - 1 < x < 1 \\ 0 otherwise \end{cases}$
(a) What is the value of $c$ ?
(b) What is the cumulative distribution function of $X$ ?

Short Answer

Expert verified

(a) The value of $c$ is $\frac{3}{4}$

(b) The cumulative distribution function of $X$ is

$F_{X} (x) = \{\begin{cases} 0 x \leq 1 \\ \frac{3}{4} x - \frac{1}{4} x^{3} + \frac{1}{2} - 1 \leq x < 1 \\ 1 x \geq 1 \end{cases}$

Step by step solution

Part (a) Step 1. Given Information.

Here, the density function of a random variable X is given as

$f (x) = \{\begin{cases} c (1 - x^{2}) - 1 < x < 1 \\ 0 otherwise \end{cases}$

Part (a) Step 2. Integration of probability density function.

We know that the probability density function of any random variable integrates to $1$ . Therefore, we have

$\int_{- \infty}^{\infty} f (x) d x = 1$

Part (a) Step 3. Solve the integration of probability density function.

$\int_{\{- \infty\}}^{1} 0 d x + \int_{\{- 1\}}^{1} c (1 - x^{2}) d x + \int_{1}^{\infty} 0 d x = 1$

$\int_{\{- 1\}}^{1} c (1 - x^{2}) d x = 1$

role="math" localid="1646480390300" $c {[x - \frac{x^{3}}{3}]}_{\{- 1\}}^{1} = 1$

$c [1 - \frac{1}{3} - (- 1) + \frac{- 1}{3}] = 1 c [\frac{3 - 1 + 3 - 1}{3}] = 1 \frac{4}{3} c = 1$

Part (a) Step 4. Determine the value of c.

$c = 1 \times \frac{3}{4} ∴ c = \frac{3}{4}$

Part (b) Step 1. Given Information.

Here, the density function of a random variable $X$ is given as

$f (x) = \{\begin{cases} c (1 - x^{2}) - 1 < x < 1 \\ 0 otherwise \end{cases}$

Part (b) Step 2. Define cumulative distribution function of X.

Cumulative distribution function of $X$ is given as

$F_{X} (x) = P |X \leq x|$

Part (b) Step 3. Solve cumulative distribution function of X.

$f_{X} (x) = \int_{\{- \infty\}}^{π} 0 d t + \int_{\{- 1\}}^{π} \frac{3}{4} (1 - t^{2}) d t$

as $c = \frac{3}{4}$

$= \frac{3}{4} {[t - \frac{t^{3}}{3}]}_{\{- 1\}}^{x} = \frac{3}{4} [x - \frac{x^{3}}{3} - (- 1) + \frac{- 1}{3}] = \frac{3}{4} [x - \frac{x^{3}}{3} + \frac{3 - 1}{3}] = \frac{3}{4} [x - \frac{x^{3}}{3} + \frac{2}{3}] = \frac{3}{4} x - \frac{x^{3}}{4} + \frac{1}{2}$

Part (b) Step 4. Write the cumulative distribution function of X.

Therefore, the cumulative distribution function of $X$ is

$F_{X} (x) = \{\begin{cases} 0 x \leq 1 \\ \frac{3}{4} x - \frac{1}{4} x^{3} + \frac{1}{2} - 1 \leq x < 1 \\ 1 x \geq 1 \end{cases}$

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Let $X$ be a random variable with probability density function
$f (x) = \{\begin{cases} c (1 - x^{2}) - 1 < x < 1 \\ 0 otherwise \end{cases}$
(a) What is the value of $c$ ?
(b) What is the cumulative distribution function of $X$ ?

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Integration of probability density function.

Part (a) Step 3. Solve the integration of probability density function.

Part (a) Step 4. Determine the value of c.

Part (b) Step 1. Given Information.

Part (b) Step 2. Define cumulative distribution function of X.

Part (b) Step 3. Solve cumulative distribution function of X.

Part (b) Step 4. Write the cumulative distribution function of X.

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Let Xbe a random variable with probability density functionf(x)=c(1-x2)−1<x<10otherwise(a) What is the value of c?(b) What is the cumulative distribution function of X?

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Integration of probability density function.

Part (a) Step 3. Solve the integration of probability density function.

Part (a) Step 4. Determine the value of c.

Part (b) Step 1. Given Information.

Part (b) Step 2. Define cumulative distribution function of X.

Part (b) Step 3. Solve cumulative distribution function of X.

Part (b) Step 4. Write the cumulative distribution function of X.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

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Calculus

Mechanics Maths

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Probability and Statistics

Study anywhere. Anytime. Across all devices.

Let $X$ be a random variable with probability density function
$f (x) = \{\begin{cases} c (1 - x^{2}) - 1 < x < 1 \\ 0 otherwise \end{cases}$
(a) What is the value of $c$ ?
(b) What is the cumulative distribution function of $X$ ?