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Let $X$ be a Bernoulli random variable with probability of success $p$. Answer the following questions.

  • (i) Derive the formulas for the mean, the variance, and the standard deviation of X.
  • (ii) Find the third and fourth moments of $X$.
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$E(e^{Xt})=E(1+Xt+X^2\frac{t^2}{2!}+\cdots)=1+E(X)t+E(X^2)\frac{t^3}{3!}+\cdots$

Now, $p_X(0)=1-p$ and $p_X(1)=p$

hence, $E(e^{Xt})=e^{0.t}p_X(0)+e^{1.t}p_X(1)=1-p +e^tp=(1-p)+p(1+t+t^2/2!+t^3/3!+\cdots)= 1+pt+p(t^2/2!)+p(t^3/3!)+\cdots$

Now, $r_{th}$ moment of $X$= coefficient of $t^r/r!$ in expansion of $E(e^{Xt})$ which is coming out to be $p$ $\forall r\in \Bbb N$

Therefore, $E(X^r)=p$ $\forall r\in \Bbb N$.

therefore, Mean=$E(X)=p$

variance= $E(X^2)-(E(X))^2=p-p^2$

Standard deviation =$\sqrt{variance}$

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$X$ has probability density function given by $$ p_X(0)=P(X=0)=(1-p),\quad p_X(1)=P(X=1)=p. $$ By knowing these probabilities, we can find the moments of any order by using the law of the unconscious statistician, i.e. $$ E[g(X)]=\sum_{x=0}^1g(x)p_X(x) $$ for any function $g$. Try this with $g(x)=x$, $g(x)=x^2$, $g(x)=x^3$ and $g(x)=x^4$, and recall that $\mathrm{Var}(X)=E[X^2]-E[X]^2$.

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