The pdf curve can be very intuitive and useful in understanding this concept.

1st example -- given U, the standard uniform RV between 0 and 1, the PDF is a square box with area under curve = 1. Now what about the derived random variable U' := 2U? Its PDF must have area under the curve = 1 but over the wider range of [0,2]. Therefore, the curve height must scale DOWN.

2nd example -- given Z, the standard normal bell curve, what about the bell curve of 2Z? It's a scaled-down, and widened bell curve, as http://en.wikipedia.org/wiki/Normal_distribution shows.

In conclusion, when we scale up a random variable by 2 to get a "derived" random variable, the density curve must scale Down by 2 (but not a simple multiply). How about the expectation? Must scale Up by 2.

## Friday, February 20, 2015

### scale up a random variable.. what about density@@

Labels: _misLabel, clarified, intuit_math, mathStat

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