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Given a probability density function find the mean, median and mode of the distribution.

Scale Invariance

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

Into the Exponential Distribution

Get into the exponential distribution through an exploration of its pdf.

Into the Normal Distribution

Age 16 to 18
Challenge Level

We can estimate the probability of selecting a negative random variable by evaluating the area under the curve in the region of negative x. Approximating this area using a triangle will give us an over-estimate of the actual probability.

Blue Curve: Pr(X< 0) $\approx$ 0.5 x 2 x 0.25 = 0.25

Red Curve: Pr(X< 0) $\approx$ 0.5 x 2 x 0.1 = 0.1

Black Curve: Pr(X< 0) $\approx$ 0.5 x 2 x 0.05 = 0.05

A normal distribution is symmetric about its mean. This allows us to estimate the mean of each distribution by inspection:

$\mu_{Blue}$ = 1

$\mu_{Red}$ = 2

$\mu_{Black}$ = 3

We know that f(x) = $\frac{1}{ \sigma \sqrt{2 \pi}} e^{\frac{-(x-\mu)^2}{2 \sigma ^2}}$

If we evaluate f(x) at x = $\mu$ the exponential will disappear ($e^0 = 1$)

We can then solve for $\sigma$

$f(\mu) = \frac{1}{ \sigma \sqrt{2 \pi}} $

$\sigma =\frac{1}{\sqrt{2 \pi} f(\mu)}$

Evaluating f($\mu$) from the curves and substituting $\mu$ into the expression we find that:

$\mu_{Blue}$ = 1, $\sigma^2_{Blue} = 1$

$\mu_{Red}$ = 2, $\sigma^2_{Red} = 2$

$\mu_{Black}$ = 3, $\sigma^2_{Black} = 3$