Chapter 4: Q169SE (page 281)

The random variable xhas a normal distribution with $μ = 75$ and $σ = 10$ . Find the following probabilities:
a. $P (x ⩽ 80)$
b. $P (x ⩾ 85)$
c. $P (70 ⩽ x ⩽ 75)$
d. $P (x > 80)$
e. $P (x = 78)$
f. $P (x ⩽ 110)$

Short Answer

Expert verified

a. $P (x ⩽ 80) =$ $0.6915$

b. $P (x ⩾ 85) =$ $0.1587$

c. $P (70 ⩽ x ⩽ 75) =$ $0.1915$

d. $P (x > 80) =$ $0.3085$

e. $P (x = 78) = 0$

f. $P (x ⩽ 110) =$ $0.9998$

Step by step solution

Given information

X is anormalrandom variable.

$μ = 75 σ = 10$

Define the probability density function

The p.d.f of X follows a normal distribution which is given by is:

$f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{1}{2} {[(x - μ) / σ]}^{2}}$

(a) Calculate Px⩽80

$\begin{array}{rcl} P (x ⩽ 80) & = & P (\frac{x - μ}{σ} ⩽ \frac{80 - 75}{10}) \\ = & P (Z ⩽ 0.5) \\ = & 0.6915 \end{array}$

Hence, $P (x ⩽ 80) =$ $0.6915$

(b) Calculate Px⩾85

$\begin{array}{rcl} P (x ⩾ 85) & = & P (\frac{x - μ}{σ} ⩾ \frac{85 - 75}{10}) \\ = & P (Z ⩾ 1) \\ = & 1 - P (Z ⩽ 1) \\ = & 1 - 0.8413 \\ = & 0.1587 \end{array}$

Hence, $P (x ⩾ 85) =$ $0.1587$

(c) Calculate P70⩽x⩽75

$\begin{array}{rcl} P (70 ⩽ x ⩽ 75) & = & P (\frac{70 - 75}{10} ⩽ \frac{x - μ}{σ} ⩽ \frac{75 - 75}{10}) \\ = & P (- 0.5 ⩽ Z ⩽ 0) \\ = & P (Z ⩽ 0) - P (Z ⩽ - 0.5) \\ = & P (Z ⩽ 0) - 1 + P (Z ⩽ 0.5) \\ = & 0.5 - 1 + 0.6915 \\ = & 0.1915 \end{array}$

Hence, $P (70 ⩽ x ⩽ 75) =$ $0.1915$

(d) Calculate Px>80

$\begin{array}{rcl} P (x > 80) & = & P (\frac{x - μ}{σ} > \frac{80 - 75}{10}) \\ = & P (Z > 0.5) \\ = & 1 - P (Z < 0.5) \\ = & 1 - 0.6915 \\ = & 0.3085 \end{array}$

Hence, $P (x > 80) = 0.3085$

(e) Calculate Px=78

$P (x = 78) = 0$ [Since, Probability of any single point is 0]

Hence, $P (x = 78) = 0$

(f) Calculate Px⩽110

$\begin{array}{rcl} P (x ⩽ 110) & = & P (\frac{x - μ}{σ} ⩽ \frac{110 - 75}{10}) \\ = & P (Z ⩽ 3.5) \\ = & 0.9998 \end{array}$

Hence, $P (x ⩽ 110) =$ $0.9998$

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Firm A			Firm B
Loss Next Year	Probabiity	Loss Next Year	Probability
0	0.01		0	0
500	0.01		200	0.01
1000	0.01		700	0.02
1500	0.02		1200	0.02
2000	0.35		1700	0.15
2500	0.3		2200	0.3
3000	0.25		2700	0.3
3500	0.02		3200	0.15
4000	0.01		3700	0.02
4500	0.01		4200	0.02
5000	0.01		4700	0.01

	Buy-Side Analysts	Sell-Side Analysts
Mean	0.85	-0.05
Standard Deviation	1.93	0.85

The random variable xhas a normal distribution with $μ = 75$ and $σ = 10$ . Find the following probabilities:
a. $P (x ⩽ 80)$
b. $P (x ⩾ 85)$
c. $P (70 ⩽ x ⩽ 75)$
d. $P (x > 80)$
e. $P (x = 78)$
f. $P (x ⩽ 110)$

Short Answer

Step by step solution

Given information

Define the probability density function

(a) Calculate Px⩽80

(b) Calculate Px⩾85

(c) Calculate P70⩽x⩽75

(d) Calculate Px>80

(e) Calculate Px=78

(f) Calculate Px⩽110

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The random variable xhas a normal distribution withμ=75and σ=10. Find the following probabilities:a.P(x⩽80)b.P(x⩾85)c.P(70⩽x⩽75)d.P(x>80)e.P(x=78)f.P(x⩽110)

Short Answer

Step by step solution

Given information

Define the probability density function

(a) Calculate Px⩽80

(b) Calculate Px⩾85

(c) Calculate P70⩽x⩽75

(d) Calculate Px>80

(e) Calculate Px=78

(f) Calculate Px⩽110

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Probability and Statistics

Mechanics Maths

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

The random variable xhas a normal distribution with $μ = 75$ and $σ = 10$ . Find the following probabilities:
a. $P (x ⩽ 80)$
b. $P (x ⩾ 85)$
c. $P (70 ⩽ x ⩽ 75)$
d. $P (x > 80)$
e. $P (x = 78)$
f. $P (x ⩽ 110)$