P(X=x)=โฉโจโงโ2kx,x=2,4,6k(x+2),x=80,otherwiseโ
Find the values โโof the probabilities
P(x=2)=2kโ
2=4k
P(x=4)=2kโ
4=8k
P(x=6)=2kโ
6=12k
P(x=8)=kโ
(8+2)=10k
a) We find the value of k from the condition
โPiโ=1
Then
4k+8k+12k+10k=1โ34k=1โk=341โ
Q.E.D
b) construct a series of probability distributions
XPโ2344โโ4348โโ63412โโ83410โโ
Let's find
P(4<xโค8)=P(6)+P(8)=3412+10โ=1711โ
Answer: P(4<xโค8)=1711โ
c) Let's find
P(2<X<4)=0 (since, by condition, P(X=x)=0 for all xโ(2;4) )
Answer: P(2<X<4)=0
d) Let's find he expected value:
M(x)=โxiโpiโ=342โ
4+4โ
8+6โ
12+8โ
10โ=1796โ
Answer: M(x)=1796โ
e) Let's find the variance:
Var(x)=M(x2)โM2(x)=344โ
4+16โ
8+36โ
12+64โ
10โโ(1796โ)2=2891120โ
Answer: Var(x)=2891120โ
f) Let's find
Var(5โ3x)=Var(5)+Var(โ3x)=0+(โ3)2Var(x)=9โ
2891120โ=28910080โ
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