# Probability Formulas

### Moments
The $r^{th}$ moment about mean is denoted by -
$$
\mu_r^\prime=E(x^r)=\begin{cases}
\sum x^r.f(x) & \text{For DRM} \\
\int_{-\infty}^{\infty} x^r.f(x) dx & \text{For CRM}
\end{cases}
$$

---

### Moment Generating Function (MGF)
$$
M_x(t)=E(tx)=\begin{cases}
\sum e^{tx}.f(x) & \text{For DRM} \\
\int_{-\infty}^{\infty} e^{tx}.f(x) dx & \text{For CRM}
\end{cases}
$$

#### Theorem
$$
\bigg[\frac{d^r M_x(t)}{dt^r}\bigg]_{t=0}=\mu_r^\prime
$$

---

## Joint Probability Distribution
$$
P((X,Y) {\Large\epsilon} A) = \int\int_A f(x,y) dx
$$

---

## Marginal Dstribution

$$
g(x)=\begin{cases}
\sum_{y}f(x,y) & \text{For DRM} \\
\int_{-\infty}^{\infty} f(x,y) dy & \text{For CRM}
\end{cases}
h(y)=\begin{cases}
\sum_{x}f(x,y) & \text{For DRM} \\
\int_{-\infty}^{\infty} f(x,y) dx & \text{For CRM}
\end{cases}
$$

---

## Conditional Distribution
$$
f(y|x)=\frac{f(x,y)}{g(x)} \hspace{1cm} \text{Provided } g(x)>0
$$
$$
f(x|y)=\frac{f(x,y)}{h(y)} \hspace{1cm} \text{Provided } h(y)>0
$$

---

## Chebyshev's Theorem
$$
P(\mu-k\sigma<X<\mu+k\sigma)\geq1-\frac{1}{k^2}
$$

---

## Poisson's Distribution
$$
P(x)=\frac{e^{-\lambda}\lambda^x}{x!}
$$

#### Mean

$$\mu=\lambda$$

#### Variance
$$\sigma=\lambda$$

---

## Skewness and Kurtosis

### Skewness
$$
\beta_1=\frac{\mu_3^2}{\mu_2^3}=\frac{1}{\lambda}
$$

$$
\beta_1
\begin{cases}
<0 & \text{Negatively skewed} \\
=0 & \text{Symmentrical} \\
\text{> } 0 & \text{Positively Skewed} \\
\end{cases}
$$

#### Kurtosis

$$
\beta_2=\frac{\mu_4}{\mu_2^2}=\frac{1}{\lambda}+3
$$

$$
\beta_2
\begin{cases}
<3 & \text{Platykurtic} \\
=3& \text{Mesokurtic} \\
\text{> } 3& \text{Leptokurtic}\\
\end{cases}
$$

## Weak Law of Large Numbers
$$
P\Bigg[\Bigg|\frac{X_1+X_2+....+X_n}{n}\Bigg|-
\mu\geq\epsilon\Bigg]\to 0 \text{ as } n\to\infty
$$

## Moment Formulaes
* $\mu_1=0$
* $\mu_2=\mu_2^\prime-(\mu_1^\prime)^2$
* $\mu_3=\mu_3^\prime-3\mu_1^\prime \mu_2^\prime+2(\mu_1^\prime)^3$
* $\mu_4=\mu_4^\prime-4\mu_1^\prime \mu_3^\prime+ 6\mu_2^\prime(\mu_1^\prime)^2-3\mu_1^\prime$

## Binomial Distribution
$$\mu = np$$
$$\sigma = npq$$