Calculating E(g(X)) for a continuous random variable · Short practice

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Recall

Continuous densities and moments — objective 1

Maths4U Original · Short practice

X has density \(f(x)=kx^{3}\) for \(0\le x\le 3\) and zero elsewhere.

Find k.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Continuous densities and moments — objective 1

Maths4U Original · Short practice

X has density \(f(x)=kx^{1}\) for \(0\le x\le 4\) and zero elsewhere.

Find k.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Method

Continuous densities and moments — objective 2

Maths4U Original · Short practice

X has density \(f(x)=kx^{1}\) for \(0\le x\le 4\) and zero elsewhere.

Find E(X).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Continuous densities and moments — objective 2

Maths4U Original · Short practice

X has density \(f(x)=kx^{2}\) for \(0\le x\le 5\) and zero elsewhere.

Find E(X).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Apply

Transformations of continuous variables — objective 3

Maths4U Original · Short practice

X has density \(\displaystyle \frac{2x}{225}\) on \(0\le x\le 15\). Define \(\displaystyle Y=X^{2}\). For \(0\le y\le 225\), \(\displaystyle F_{Y}(y)= P(X\le \sqrt{y})= \frac{y}{225}\).

Find Var(Y).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Transformations of continuous variables — objective 3

Maths4U Original · Short practice

X has density \(\displaystyle \frac{2x}{256}\) on \(0\le x\le 16\). Define \(\displaystyle Y=X^{2}\). For \(0\le y\le 256\), \(\displaystyle F_{Y}(y)= P(X\le \sqrt{y})= \frac{y}{256}\).

Find Var(Y).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Check understanding

Transformations of continuous variables — objective 4

Maths4U Original · Short practice

X has density \(\displaystyle \frac{2x}{256}\) on \(0\le x\le 16\). Define \(\displaystyle Y=X^{2}\). For \(0\le y\le 256\), \(\displaystyle F_{Y}(y)= P(X\le \sqrt{y})= \frac{y}{256}\).

Find \(\displaystyle P(Y\le 1)\).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Transformations of continuous variables — objective 4

Maths4U Original · Short practice

X has density \(\displaystyle \frac{2x}{289}\) on \(0\le x\le 17\). Define \(\displaystyle Y=X^{2}\). For \(0\le y\le 289\), \(\displaystyle F_{Y}(y)= P(X\le \sqrt{y})= \frac{y}{289}\).

Find \(\displaystyle P(Y\le 1)\).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

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