Random Variables (expectation value, var(x) and standard deviation

Consider the random variables X

x 1 | 2 | 3 P(x) 0.3 |0.5 |0.2

Find the distribution, mean µY , variance and standard deviation σY of the random variable Y = Φ(X) where (a) Φ(x) = x^3 (b) Φ(x) = 2^x.

My question is. When i will calculate the expectation value..will i have to do it based on the table above which is the distribution of x, or i have to calculate it based on the table of x^3?

• What is the distribution of $X$? – Jonas Dec 28 '18 at 14:03
• i have added it – Anastasia Kyriakou Dec 28 '18 at 14:07

1 Answer

Guide:

Apply: $$\mathbb Ef(X)=\sum_x f(x)P(X=x)$$

In your case just a sum of $$3$$ terms, so calculators are not needed.

This in order to find $$\mathbb E\Phi(X)$$ and $$\mathbb E\Phi(X)^2$$.

Then you can also find variance and standard deviation.

• Yes but when i will calculate the expectation value, will i do : 1*0.3+2*0.5+3*0.2 or i will do 1*0.3+4*0.5+9*0.2 because it says Φ(x) = x^3 ? – Anastasia Kyriakou Dec 28 '18 at 14:14
• The first calculation in your comment is a calculation of $\mathbb EX=\sum_xxP(X=x)$ and the second is a calculation of $\mathbb EX^2=\sum_xx^2P(X=x)$. Since $\Phi(x)=x^3$ you should go for $\mathbb E\Phi(X)=\mathbb EX^3=\sum_xx^3P(X=x)$. – drhab Dec 28 '18 at 14:17
• Okay and if i use only the table of x and x^2 of course to find the variance, where will i use the Φ(x) = x^3 ? – Anastasia Kyriakou Dec 28 '18 at 14:19
• $\mathsf{Var}(\Phi(X))=\mathbb E\Phi(X)^2-(\mathbb E\Phi(X))^2$. You already found $\mathbb E\Phi(X)$ so it remains to find $\mathbb E\Phi(X)^2=\mathbb EX^6$.This with the same method. – drhab Dec 28 '18 at 14:21
• yes i know how to calculate.but why the question says for Φ(x) = x^3? what does change? – Anastasia Kyriakou Dec 28 '18 at 14:24