Concepts and reason
The concept is based on the joint probability distribution function. A joint probability is a statistical measure where the occurrences of two events are together. It is also called the intersection of two or more events. The marginal probability distribution of the random variable gives the distribution of the random variable without considering the occurrence of the other variable. Two events are said to be independent if the occurrence of one event does not affect the occurrence of another event.
Fundamentals
(b.2)
Substitute the obtained marginal probability density functions and the given joint probability density function under the independence condition to find out if the two lifetimes are independent.
Since the product of the marginal probability density function is not equal to the given joint probability density function, the two lifetimes are not independent.
The two random variables representing the lifetimes of the components are not independent.
Since the joint probability distribution of the random variables is not presented as the product of two marginal distributions, the random variables are not independent of each other, i.e. the occurrence of one event affects the occurrence of another.[presentedintheformoftheproductoftwomarginaldistributionsotherandomvariablearenotindependentofeachotherthatistheoccurrenceofoneeventeffectstheoccurrenceofother[présentéssouslaformeduproduitdedeuxdistributionsmarginalesd’autresvariablesandomsontpasindépendantesl’unedel’autrequelasurvenanced’unévénementaffectelasurvenanced’autres[presentedintheformoftheproductoftwomarginaldistributionsotherandomvariablearenotindependentofeachotherthatistheoccurrenceofoneeventeffectstheoccurrenceofother
to check the independence of the random variables because it is an error. use the correct formula 
.
to find the probability that the lifetime of at least one component exceeds 3.(vs)
The probability that the lifetime of at least one component exceeds 3 is given by:
part c
The probability that the lifetime of at least one component exceeds 3 is 0.30.
The probability indicates that there is a 30% chance that the lifetime of at least one of the components exceeds 3.
The probability that the lifetime X of the first component exceeds 3 is 0.05.
The marginal density function needed to X and s is obtained as:
The two random variables representing the lifetimes of the components are not independent.
The probability that the lifetime of at least one component exceeds 3 is 0.30.
The probability that the lifetime X of the first component exceeds 3 is 0.05.
The marginal density function needed to X and s is obtained as:
The two random variables representing the lifetimes of the components are not independent.
The probability that the lifetime of at least one component exceeds 3 is 0.30.
f(x,y)
P(x,ye A)=P(like X ,csy>
:(x)= S(x,y) dy for –0
effect (x)
f(y)
f(x,y)= f(x)*f, (y) for all x and y
Plas X sb,csYsd)= fjs(x,y) dy de
X and Y
s(x,y)= {0x and (1+y). x20 and y20 otherwise
3 to 60
sa Icro-2-1) = Pp 8? (v)_* }={E
Plas X sb,csYsd)= fjs(x,y) dy de
Plas X sb,csYsd)= fjs(x,y) dy de
ap(4x) =(x)
1,(v)=JS(x,y) of
X and Y
$(x)=}(x,y) dy to -0
J,(v)=(x-2) do CHOPI–0 (y+1)x+1)e-xx-x7 y +2y+1 (1+y) ?
X and Y
S.(=)=f5(x,y) of
s»(y)=ff(x,y) dy
X and Y
ap(4x) =(x)
1,(v)=JS(x,y) of
f(x,y)= f(x)*f,(y)
f(x,y) = fx(x)fy(y) te **» *fe ») and
f(x,y) = f(x)+f, (y)
f(x,y)= f(x)*f,(y)
Plas X sb,csYsd)= fjs(x,y) dy de
P(X> 3 or Y > 3)=1– P(X,Y 53) =– ( – *»I det =1-}(e**(* =1)
P(X>3 or Y>3)=1-LETA = 1-0.700 =0.3000
1x(x)=e* x20 else y20 S, (y)={(1+y) ? 10 otherwise
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