Chapter 4: Q164S (page 281)

Assume that xis a random variable best described by a uniform distribution with $c = 10$ and $d = 90$ .
a. Find $f (x)$ .
b. Find the mean and standard deviation of x.
c. Graph the probability distribution for xand locate its mean and theintervalon the graph.
d. Find $P (x \leq 60)$ .
e. Find $P (x \geq 90)$ .
f. Find $P (x \leq 80)$ .
g. Find $P (μ - σ \leq x \leq μ + σ)$ .
h. Find $P (x > 75)$ .

Short Answer

Expert verified

a. $f (x) = \frac{1}{80}; 10 \leq x \leq 90$

b. The mean is $μ = 50$ and the standard deviation is $σ = 23.094$ .

c. The graph is a symmetrical curve.

d. $P (x \leq 60) = 0.625$

e. $P (x \geq 90) = 0$

f. $P (x \leq 80) = 0.875$

g. $P (μ - σ < x < μ + σ) = 0.57735$

h. $P (x > 75) = 0.57735$

Step by step solution

Given information

X is auniformrandom variable.

$c = 10 d = 90$

Calculate f(x)

The p.d.f of X is

$f (x) = \frac{1}{d - c}; c \leq x \leq d = \frac{1}{90 - 10} = \frac{1}{80}; 10 \leq x \leq 90$

Hence, $f (x) = \frac{1}{80}; 10 \leq x \leq 90$ .

Calculate mean and standard deviation

Mean= $μ$

$μ = \frac{c + d}{2} = \frac{(10 + 90)}{2} = \frac{100}{2} = 50$

Standard deviation= $σ$

$σ = \frac{d - c}{\sqrt{12}} = \frac{90 - 10}{\sqrt{12}} = \frac{80}{\sqrt{12}} = 23.094$

Hence, the mean is $μ = 50$ and standard deviation is $σ = 23.094$ .

Plot the graph

$μ \pm 2 σ = 50 \pm (2 \times 23.094) 50 \pm 46.188 = (3.812, 96.188)$

Calculate P(x≤60)

$P (x \leq 60) = \int_{10}^{60} f (x) d x = \int_{10}^{60} \frac{1}{80} d x = \frac{1}{80} \int_{10}^{60} d x = \frac{1}{80} \times 50 = \frac{5}{8} = 0.625$

Hence, $P (x \leq 60) = 0.625$ .

Calculate P(x≥90)

$P (x \geq 90) = 1 - P (x < 90) = 1 - \int_{10}^{90} f (x) d x = 1 - \int_{10}^{90} \frac{1}{80} d x = 1 - (\frac{1}{80} \times 80) = 1 - 1 = 0$

Hence, $P (x \geq 90) = 0$ .

Calculate P(x≤80)

$P (x \leq 80) = 1 - \int_{80}^{90} f (x) d x = 1 - \int_{80}^{90} \frac{1}{80} d x = 1 - (\frac{1}{80} \times 10) = 1 - \frac{1}{8} = 0.875$

Hence, $P (x \leq 80) = 0.875$ .

Calculate P(μ-σ≤x≤μ+σ)

$P (μ - σ < x < μ + σ) = \int_{μ - σ}^{μ + σ} f (x) d x = \int_{26.906}^{73.094} \frac{1}{80} d x = \frac{1}{80} \times 46.188 = 0.57735$

Hence,role="math" localid="1660281305747" $P (μ - σ < x < μ + σ) = 0.57735$ .

Calculate

$P (x > 75) = \int_{75}^{90} f (x) d x = \int_{75}^{90} \frac{1}{80} d x = \frac{1}{80} \times 15 = \frac{3}{16} = 0.57735$

Hence, $P (x > 75) = 0.57735$ .

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Short Answer

Step by step solution

Given information

Calculate f(x)

Calculate mean and standard deviation

Plot the graph

Calculate P(x≤60)

Calculate P(x≥90)

Calculate P(x≤80)

Calculate P(μ-σ≤x≤μ+σ)

Calculate

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