Topic B6 — Continuous Distributions

Table of contents
  1. Topic B6 — Continuous Distributions
    1. Probability density function f(x)f(x)
    2. Cumulative distribution function (CDF) F(x)=P(X<x)F(x) = P(X<x)
    3. Uniform distribution U(a,b)U(a,b)
    4. Compute expected value
    5. Compute CDF
    6. Normal distribution XN(μ,σ)X \sim N(\mu,\sigma)
    7. Symmetry around the mean
    8. Changes in μ\mu vs changes in σ\sigma
    9. Standard normal distribution ZN(0,1)Z \sim N(0,1)
    10. Standardization of normal random variables Z=Xμσ Z = \frac{X-\mu}{\sigma}

Probability density function f(x)f(x)

The area under the function f(x)f(x) over all values of xx equals 1. That is, the integral equals 1.

No probability attached to a specific value.

Cumulative distribution function (CDF) F(x)=P(X<x)F(x) = P(X<x)

Probabilty that random variable XX is below xx. S-shaped function of x.

Proba that random var is between a and b P(a<X<b)=F(b)F(a)=abf(u)duP(a < X < b) = F(b) - F(a) = \int^b_a f(u)du

Uniform distribution U(a,b)U(a,b)

All the values have the same probability of happening. Computed as 1ba \frac{1}{b-a} (height of the fnct) as the proba of axb a \leq x \leq b , 0 for anything else.

Compute expected value

The middle of the range, a+b2 \frac{a+b}{2}

Variance : (ba)212 \frac{(b-a)^2}{12}

Compute CDF

Uniform distrib of tips between 50 and 125.

Height? 1ba \frac{1}{b-a}

Proba of having between max and 100? 1ba(max100) \frac{1}{b-a} (max - 100)

Normal distribution XN(μ,σ)X \sim N(\mu,\sigma)

$f(x)= \frac{1}{\sqrt{2\pi \sigma^2}} exp \left{ \frac{(x-\mu)^2}{2\sigma^2} \right} $ don’t need to know this by heart!

It is completely described by μ\mu and σ\sigma

Symmetry around the mean

There is 50% probability that XμX \geq \mu and XμX \leq \mu

Changes in μ\mu vs changes in σ\sigma

Change in μ\mu: moves on the x-axis

Changes in σ2\sigma^2: bigger σ2\sigma^2 –> lower crest, thicker tails.

Standard normal distribution ZN(0,1)Z \sim N(0,1)

Normal with mean = 0, std dev = 1

Standardization of normal random variables Z=Xμσ Z = \frac{X-\mu}{\sigma}

If we got a normally distributed variable, we can standardize it using this formula Z=Xμσ Z = \frac{X-\mu}{\sigma} . That is, apply this formula to all the observations zz, also called z-scores. Std Dev becomes 1 (as seen above)

Inverse transformation, simple algebra. You want XX isolated on the left of the = sign.

Tables help you and give you P(Z<z)P(Z < z). Look for the first decimal down the first row, and second decimal across the columns

The probability for negative values: P(Z<z)=P(Z>z)=1P(Z<z)P(Z < -z) = P(Z > z) = 1-P(Z<z)