Question

A random variable x takes four possible values, 0, 2, a, and b, where a and b are unknown numbers. if pr[x = 0] = 0.2, pr[x=2] = 0.4, pr[x = a] = 0.3, e[x] = 2.8 and it is the case that a + b = 6, what is the missing value of a ?

a.-1.0
b.5.0
c.7.0
d.cannot be determined without more information
e.none of the above

Answer 1

`\displaystyle\sum_{x\in\{0,2,a,b\}}\mathbb P(X=x)=1`

`\mathbb P(X=0)+\mathbb P(X=2)+\mathbb P(X=a)+\mathbb P(X=b)=1`

`0.2+0.4+0.3+\mathbb P(X=b)=1`

`\implies\mathbb P(X=b)=0.1`


`\mathbb E(X)=\displaystyle\sum_{x\in\{0,2,a,b\}}x\mathbb P(X=x)=2.8`

`0\mathbb P(X=0)+2\mathbb P(X=2)+a\mathbb P(X=a)+b\mathbb P(X=b)=2.8`

`0.8+0.3a+0.1b=2.8`

`0.3a+0.1b=2`


`a+b=6\implies0.3a+0.1b=0.2a+0.1(6)=2`

`0.2a+0.6=2\implies0.2a=1.4\implies a=7`