07 - Continuous Random Variables

Continuous Random Variables

PD vs CD

• PD: probability density: calculate at specific point $P(X=k)$ → f(a)
• CD: cumulative distribution: calculate within a range $P(a\le X\le b)$ → F(a)
• pdf = deri cdf

Definition

• X is a continuous random variable (CRV) if $P(X\in B)={\int }_{B}f(x)dx$ $f$: nonnegative function $x\in (-\infty ,\infty )$ $B$ is a set of real numbers
• function $f$, probability density function of the RV $X$, must satisfy ${\int }_{-\infty }^{\infty }f(x)=1$
• We have $P(a\le X\le b)={\int }_a^{b}f(x)dx$ $P(X=a)={\int }_a^{a}f(x)dx=0$ Hence $P(X\le a)=P(X<a)$

Expected Value

• If X is a CRV having f(x), $E[X]={\int }_{-\infty }^{\infty }x\cdot f(x)dx$

Proposition

• X a CRV with f(x), for real-valued g $E[g(X)]={\int }_{-\infty }^{\infty }g(x)\cdot f(x)dx$

Corollary

• a, b const: $E[aX+b]=aE[X]+b$

Variance & standard deviation

• 

Uniform distribution

• 
• Probability of uniform distribution $P(x_1<X<x_2)=\frac{x_2-x_1}{b-a}$

Normal distribution

Standard Normal Distribution

• area of z-value is to the left
• determine value of z on z-table: z = 1.56 → 1.5 in rows - vertical, 0.06 in columns - horizontal

Normal Approximation to Binomial Distribution

• trials are large, evaluate bino is difficult satisfy $np\ge 5$, $n(1-p)\ge 5$ normal distribution can be used to approx

• how to use normal approx to bino

  • satisfy $np\ge 5$, $n(1-p)\ge 5$, set $\mu =np,\sigma =\sqrt{np(1-p)}$
  • set $\mu =np,\sigma =\sqrt{np(1-p)}$ in definition of the normal curve
  • Bino is discrete integer-valued RV
  • Normal is continuous RV
  • continuity correction: $P(X=i)$ →$P(i-0.5<X<i+0.5)$ $np\ge 5$, $n(1-p)\ge 5$, set $\mu =np,\sigma =\sqrt{np(1-p)}$

• DeMoivre-Laplace limit theorem

  • normal approx to bino is result of this theorem, which is special case of central limit theorem

Exponential Random Variables

• Note
  • $X<x$: approx with normal approx bino
  • $X=x$: approx with poisson