[Probability] Discrete, Continuous, Joint Probability Distributions - Examples

Example 3.8 

A shipment of 20 similar laptop computers to a retail outlet contains 3 that are defective. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives.

 

Solution : 

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Example 3.11

Suppose that the error in the reaction temperature, in °C, for a controlled laboratory experiment is a continuous random variable X having the probability density function.

(a) Verify that f(x) is a density function

(b) Find P(0<X≤1).

 

Solution : 

 

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Example 3.12

For the density function of Example 3.11, find F (x), and use it to evaluate P (0 < X ≤ 1).

 

Solution : 

①x < -1

F(x) = 0

 

②-1 ≤x≤2

((x^3)/9)+(1/9)

 

③x>2

F(x) = 1

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Example 3.14

Two ballpoint pens are selected at random from a box that contains 3 blue pens, 2 red pens, and 3 green pens. If X is the number of blue pens selected and Y is the number of red pens selected, find

(a) the joint probability function f(x,y),
(b) P[(X,Y)∈A],whereAistheregion{(x,y)|x+y≤1}.

 

Solution :

(a)

 

(b)

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Example 3.15

A privately owned business operates both a drive-in facility and a walk-in facility. On a randomly selected day, let X and Y , respectively, be the proportions of the time that the drive-in and the walk-in facilities are in use, and suppose that the joint density function of these random variables is

Solution :