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Chapter 4: Q165S (page 281)

Find the following probabilities for the standard normal

random variable z:

a.P(z≤2.1)

b.P(z≥2.1)

c.P(z≥-1.65)

d.P(-2.13≤z≤-.41)

e.P(-1.45≤z≤2.15)

f.P(z≤-1.43)

Short Answer

Expert verified

a. P(z≤2.1)=0.9821

b. P(z≥2.1)=0.0179.

c. P(z≥-1.65)=0.9505

d..P(-2.13≤z≤-.41)=0.3243

e.P(-1.45≤z≤2.15)=0.9107

f. P(z≤-1.43)=0.0764

Step by step solution

01

Given information

Z is a standard normal random variable.

02

Calculate P(z≤2.1)

a.

In case of the standard normal random variable z, we can find probability Pz≤2.1as follows:

Pz≤21=0.9821

Hence, Pz≤21=0.9821.

03

Calculate P(z≥2.1)

b.

In case of the standard normal random variable z, we can find probabilityPz≤-1.43 as follows:

Pz≤-1.43=1-Pz≤-1.43=1-0.9236=0.0764

Hence,Pz≤2.1-0.0179 .

04

Calculate P(z≥-1.65)

c.

WhenPz≥-1.65,

Pz≥-1.65=Pz≥-1.65=0.9505

Hence, Pz≥-1.65=0.9505.

05

Calculate P(-2.13≤z≤-.41)

d.

P(-2.13≤z≤-.41)=Pz≤-.41-Pz≤-2.13=1-Pz≤-.41-1+Pz≤2.13=1-0.6591-1+0.9834=0.3243

Hence,P(-2.13≤z≤-.41)=0.3243.

06

Calculate P(-1.45≤z≤2.15)

e.

P-1.45≤2.15=Pz≤2.15-Pz≤-1.15=Pz≤2.15-1+Pz≤1.15=0.9842-1+0.9265=0.9107

Hence, P-1.45≤z≤2.15=0.9107.

07

Calculate P(z≤-1.43)

f.

Pz≤-1.43=1-Pz≤1.43=1-0.9236=0.0764

Hence, Pz≤-1.43=0.0764.

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Most popular questions from this chapter

Explain why each of the following is or is not a valid probability distribution for a discrete random variable x:

a.

x0123
P(x).1.3.3.2

b.

x-2-10
P(x).25.50.25

c.

x4920
P(x)-3.4.3

d.

x2356
P(x).15.15.45.35

The random variable x has a normal distribution with μ=1000 and σ=10.

a. Find the probability that x assumes a value more than 2 standard deviations from its mean. More than 3 standard deviations from .μ

b. Find the probability that x assumes a value within 1 standard deviation of its mean. Within 2 standard deviations of μ.

c. Find the value of x that represents the 80th percentile of this distribution. The 10th percentile.

If x is a binomial random variable, calculate , , and for each of the following:

  1. n = 25, p = .5
  2. n = 80, p = .2
  3. n = 100, p = .6
  4. n = 70, p = .9
  5. n = 60, p = .8
  6. n = 1000, p = .04

Types of finance random variables. Security analysts are professionals who devote full-time efforts to evaluating the investment worth of a narrow list of stocks. The following variables are of interest to security analysts. Which are discrete and which are continuous random variables?

a. The closing price of a particular stock on the New York Stock Exchange.

b. The number of shares of a particular stock that are traded each business day.

c. The quarterly earnings of a particular firm.

d. The percentage change in earnings between last year and this year for a particular firm.

e. The number of new products introduced per year by a firm.

f. The time until a pharmaceutical company gains approval from the U.S. Food and Drug Administration to market a new drug.

Suppose x is a binomial random variable with n = 3 and p = .3.

  1. Calculate the value of p(x),role="math" localid="1653657859012" x=0,1,2,3,using the formula for a binomial probability distribution.
  2. Using your answers to part a, give the probability distribution for x in tabular form.
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