Chapter 5: Q:5.3 (page 217)

For some constant c, the random variable X has the probability density function:
$f (x) = \{\begin{cases} c x^{4} 0 < x < 2 \\ 0 otherwise \end{cases}$
Find
$E [X]$ and
$V a r (X)$

Short Answer

Expert verified

a. The value of $E [X] = \frac{5}{3}$

b. $Va r (X) = \frac{5}{63}$

Step by step solution

Step:1 Given Information (Part a)

The random variable X has the probability density function

$f (x) = \{\begin{cases} c x^{4} 0 < x < 2 \\ 0 otherwise \end{cases}$

We have to find $E [X]$

Definition (part a)

The Probability Density Function (PDF) is a probability function that represents the density of a continuous random variable that falls inside a given range of values.

Step:3 Explanation (Part a)

Let's start by determining the c constant. There must be one,

$1 = \int_{ℝ} f (x) d x = \int_{0}^{2} c x^{4} d x = {c (\frac{x^{5}}{5})|}_{0}^{2} = c \cdot \frac{32}{5}$

which implies that $c = \frac{5}{32}$

(a) The mean value is

$E (X) = \int_{ℝ} x f (x) d x = \int_{0}^{2} x \cdot \frac{5}{32} x^{4} d x = \frac{5}{32} \int_{0}^{2} x^{5} d x = {\frac{5}{32} (\frac{x^{6}}{6})|}_{0}^{2} = \frac{5}{32} \times \frac{64}{6} = \frac{5}{3}$

Explanation (Part b)

According to the information, We observe that:

$E (X^{2}) = \int_{0}^{2} x^{2} \cdot \frac{5 x^{4}}{32} d x = \frac{5}{32} {(\frac{x^{7}}{7})}_{0}^{2}$

$\frac{5}{32} (2^{7} - 0) = \frac{5}{32 \times 7} \times 128 = \frac{20}{7}$

Therefore $V a r (X)$ :

localid="1650014593098" $V a r (X) = E (x^{2}) - (E (x))^{2}$

localid="1649597273687" $\Rightarrow V a r (X) = \frac{20}{7} - {(\frac{5}{3})}^{2} = \frac{5}{63}$

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For some constant c, the random variable X has the probability density function:
$f (x) = \{\begin{cases} c x^{4} 0 < x < 2 \\ 0 otherwise \end{cases}$
Find
$E [X]$ and
$V a r (X)$

Short Answer

Step by step solution

Step:1 Given Information (Part a)

Definition (part a)

Step:3 Explanation (Part a)

Explanation (Part b)

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For some constant c, the random variable X has the probability density function:f(x)=cx4 0<x<20 otherwiseFindE[X]andVar(X)

Short Answer

Step by step solution

Step:1 Given Information (Part a)

Definition (part a)

Step:3 Explanation (Part a)

Explanation (Part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

For some constant c, the random variable X has the probability density function:
$f (x) = \{\begin{cases} c x^{4} 0 < x < 2 \\ 0 otherwise \end{cases}$
Find
$E [X]$ and
$V a r (X)$