Question

When an automobile is stopped by a roving safety patrol, each tire is checked for tire wear, and each headlight is checked to see whether it is properly aimed. LetX denote the number of headlights that need adjustment, and let Y denote the number of defective tires.(a) If X and Y are independent with pX(0) = 0.5, pX(1) = 0.3, pX(2) = 0.2, and pY(0) = 0.6, pY(1) = 0.2, pY(2) = pY(3) = 0.05, pY(4) = 0.1, display the joint pmfof (X, Y) in a joint probability table.
y
p(x, y)01234
x

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0

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1

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2

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(b) Compute P(X ≤ 1 and Y ≤ 1) from the joint probability table.
P(X ≤ 1 and Y ≤ 1) =

Does P(X ≤ 1 and Y ≤ 1) equal the product P(X ≤ 1) · P(Y ≤ 1)?
Yes No

(c) What is P(X + Y = 0) (the probability of no violations)?
P(X + Y = 0) =

(d) Compute P(X + Y ≤ 1).
P(X + Y ≤ 1) =

Answer 1

Let X denotes the number of headlights that need adjustment, and let Y denote the number of defective tires.

a)

Let X and Y be independent with

And

If X and Y are independent random variables then the joint probabilities are obtained by using the following formula,

If then

If then

Similarly,

If then

Therefore, the joint probability table is shown below:

				y
	P(x,y)	0	1	2	3	4
	0	0.3	0.05	0.025	0.025	0.1
x	1	0.18	0.03	0.015	0.015	0.06
	2	0.12	0.02	0.01	0.01	0.04

b)

We find the value of

Where as

Therefore,

c)

We find

Here is possible only when

From the joint probability tables,

d)

We compute the value of is

Therefore, the value of is 0.53.