If X has distribution function F, what is the distribution function of the random variable $\alpha X + \beta$, where $\alpha$ and $\beta$ are constants?
Solution
Let $F(x)=P(X\leq x)$ be the distribution function of random variable $X$.
Let $Y = \alpha X +\beta$. Let the distribution function of $Y$ be $G(y)$.
$$ \begin{aligned} G(y) &= P(Y\leq y)\\ &= P(\alpha X + \beta\leq y)\\ &= P(\alpha X \leq y-\beta)\\ &= P( X \leq \frac{y-\beta}{\alpha})\\ &= F\bigg(\frac{y-\beta}{\alpha}\bigg) \end{aligned} $$
Thus the distribution function of $\alpha X+ \beta$ is
$F\bigg(\dfrac{y-\beta}{\alpha}\bigg)$.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
