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have adiscreet random variable and cumulative distribution function ,how do i get the probability function,mean ,variance x 1 2 3 4 5 6 7 8 F(x) 0.1 0.2 0.25 0.4 0.5 0.6 0.75 1

determine the probability function,f(x)=p(X=x),of x(2). calculate E(X) VAR(X) (4) GIVEN THAT Y=2x+3,find the mean and variance of y(2)

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Hint: if $F$ is the cumulative density function of $X$ on $\Omega=\{1,\dots,n\}$ and $f$ its probability mass function, then $$ \forall k \in \Omega,\quad F(k) = \sum_{i=1}^k f(i) $$ (This is a special case of $F(x)=\mathbb{P}\!\left\{X\in(-\infty, x]\right\}$.) Now, express $f(k)$ as $F(k)-F(k-1)$...

For the expectation $\mathbb{E}X$, by definition $$ \mathbb{E}X = \sum_{k=1}^n k \cdot f(k) $$ and for the variance, also by definition $$ \operatorname{Var}X = \mathbb{E}[X^2] - ( \mathbb{E}X )^2 = \mathbb{E}X = \sum_{k=1}^n k^2\cdot f(k) - \left(\sum_{k=1}^n k \cdot f(k)\right)^2. $$

Finally, for $Y$, recall that the expectation is a linear operator: for all $a,b\in\mathbb{R}$, $$ \mathbb{E}[aX+b] = a\mathbb{E}[X] + b. $$ For the variance, you can either use the definition above (with $Y$ instead of $X$) to compute it directly, or use the following two properties of the variance:

  • it is translation-invariant: for all $b\in\mathbb{R}$, $\operatorname{Var}(X+b) = \operatorname{Var}X$;
  • it scales quadraticly: for all $a\in\mathbb{R}$, $\operatorname{Var}(aX) = a^2\operatorname{Var}X$.
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