4.6 - Continuous Random Variables

Probability = area under probability density curve
Probability density curves are:

Non-negative
Total area = 1

P (a < x < b) = P (a \leq x \leq b) = P (a < x \leq b) = P (a \leq x < b)

Example Curves

For all values between $a$ and $b$ the probability densities are the same
Uniform c.r.v

\begin{aligned} Area = 1 & = length \times height \\ 1 & = (B - A) \times height \\ heigth & = \frac{1}{B - A} \end{aligned}

So to find a probability of event $P (U > C)$

We can do

P (U > C) = height \times (C - B) = \frac{1}{B - A} \times (C - B)

\begin{aligned} 1 = & \frac{1}{2} \times (11 - 7) \times h \\ h = & \frac{1}{2} \end{aligned}

$μ = \frac{4 + 7}{2} = 5.5$

if we convert it into a $z$ distribution

$μ = 0$
$σ = 1$
$P (z < P) = norm.s.dist (P, 1)$
$P (X < a) = blue area = norm.dist (a, μ, σ, 1) = red area = norm.s.dist (a^{'}, 1)$