Chapter 8 Continuous Probability Distributions

Question 1

Continuous probability distributions

Answer 1

Used to calculate the probability associated with an interval variable

Can use a continuous distribution to approximate a discrete distribution

Question 2

Continuous random variable

Answer 2

Random variable that can assume an uncountable number of values

Probability of each individual value is virtually 0, must determine probability for range

Sum of all probabilities still = 1

Question 3

Probability density function requirements

Answer 3

F(x) with range a<=x <=b

• f(x)>=0 for all x between a and b
• total area under the curve f(x) between a and b is 1

Probability that a variable is in the given range is the area of the curve under the function for that range

Question 4

Uniform probability distribution

Answer 4

Aka rectangular probability distribution

f(x) = 1/(b-a)

Where a<=x<=b

Question 5

Probability for interval with uniform distribution

Answer 5

Area of rectangle whose base is x2-x1 and whose height is 1/(b-a)

Question 6

Normal density function

Answer 6

Probability density function for a normal random variable

f(x)= 1/(standard deviationsqrt of 2π) * e^ -1/2((x-mean)/standard deviation)^2

For values of x between - infinity and infinity

Bell curve

Question 7

Normal distribution

Answer 7

Function creates a symmetric curve around the mean

Described by the mean and the standard deviation

Increasing the mean shifts the curve right, decreasing the mean shifts the curve left (along the x axis)

Increasing the standard deviation widens the curve, decreasing narrows it

Question 8

Calculating normal probabilities

Answer 8

The probability that a normal random variable falls into an interval is the area of the interval under the curve

Instead of doing for any given curve rather standardize the normal random variable and then use the table for cumulative area/probability to that variable

Question 9

Standard normal random variable

Answer 9

For any X (normal random variable)

Z(standard normal random variable)= (X - mean)/ standard deviation

The area under the standard normal distribution is 1 so the cumulative probabilities for each point are known values

Value of Z is the number of standard deviations x is from mean

Question 10

To determine probability of random variable greater than a value of z

Answer 10

Use complement Rule

P(Z>#) = 1 - P(Z

Question 11

Probability z is between two numbers

Answer 11

Cumulative probability of higher number less cumulative probability of lower number

Question 12

Limits of area under standard deviation

Answer 12

P(Z3.10) approx 0

Question 13

Finding the value of Z given a particular probability

Answer 13

Use Z(subscript A) to represent the value of Z such that the area to it’s right under the standard normal curve is A

P(Z>ZA)= A

First must find the compliment of A (1-A)
Then find that value on the table and figure out what value of Z is associated with it

If between two z values must find average of those two

Question 14

Finding value of - ZA

Answer 14

Simply the negative value of ZA.

Because the standard normal curve is symmetric around 0

Question 15

Finding value of X given area ZA

Answer 15

Find complement 1- A
Find corresponding value of Z using table

Use standardizing equation to solve for x

Z(known) = (x- mean)/standard deviation

Question 16

ZA percentile

Answer 16

For a value of ZA

That value of Z marks the 100*(1-A)th percentile

Question 17

To compute cumulative normal distribution probabilities

Answer 17

=NORMDIST(X, mean, standard deviation, TRUE)

will never use false because it will give the meaningless non-cumulative probability

Question 18

To find the cumulative probability of Z for a standard normal probability

Answer 18

= NORMDIST(Z)

Question 19

To find value of Z for a given A (standard normal distribution)

Answer 19

=NORMSINV(1-A)

Question 20

To find the value of x for a given probability A (normal distribution)

Answer 20

=NORMINV((1-A), mean. Standard deviation)

Question 21

Exponential probability density function

Answer 21

Probability density of random variable x represented by:

F(x)=lambdae^(-lambdax)

Where X >=0

And mean = standard deviation = 1/lambda

F(0)=lambda
F(x) approaches 0 as x approaches infinity

Useful for time probability functions

Question 22

Probability associated with an exponential random variable

Answer 22

If X is an exponential random variable

P(X>x) = e^(-lambda*x)
P(X

Question 23

Cumulative exponential probability in excel

Answer 23

=EXPONDIST(X, lambda, True)