There are two trails near Leo's house that he runs regularly, a short loop and a long loop. Last week, he ran 6 short loops and 5 long loops, for a total of 75 kilometers. This week, he
ran 4 short loops and 4 long loops, covering a total of 56 kilometers. What is the length of
each loop?
kilometers, and the long loop has a length of
The short loop has a length of
kilometers.

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Ответ:

kimberlynassar1962

04.11.2021

Mathematics

E[X] = 1.2

E[X2] = 1.74

Step-by-step explanation:

If X is the number of games shankar wins, the sample space for X is [0,1,2], as he can win none, on or the two games.

Let:

Pw1: probability of winning the first game

Pw2: probability of winning the second game

The probability of winning none of the games (X=0) is:

P(X=0)=(1-Pw1)(1-Pw2)=(1-0.3)(1-0.9)=0.7*0.1=0.07

The probability of winning only one of the games (X=1) is:

$P(X=1)=(1-Pw1)(Pw2)+(Pw1)(1-Pw2)=(1-0.3)(0.9)+(0.3)(1-0.9)\\\\P(X=1)=0.7*0.9+0.3*0.1=0.63+0.03=0.66$

The probability of winning both games (X=2) is:

P(X=2)=Pw1*Pw2=0.3*0.9=0.27

The expected value E(X) is:

$E(X)=\sum p_iX_i=0.07*0+0.66*1+0.27*2=0+0.66+0.54=1.2$

The expected value of the square of X, E(X^2) can be calculated as:

$E(X^2)=\sum p_iX_i^2=0.07*0^2+0.66*1^2+0.27*2^2\\\\E(X^2)=0+0.66*1+0.27*4=0.66+1.08=1.74$

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