Important Formulas for Probability Distributions

Discrete Probability Distributions

A discrete random variable can take on specific, countable values. This often happens in situations where we can list out all possible outcomes. Here are some key formulas to remember for discrete probability distributions:

1. Probability Mass Function (PMF):
   For a discrete random variable called $X$, the PMF is written as $P(X = x)$. It tells us the chance of $X$ having a particular value, $x$. Here are some important points about PMF:

   • The probability $P(X = x)$ is always greater than or equal to 0.
   • If you add up the probabilities of all possible values of $x$, you should get 1.

2. Cumulative Distribution Function (CDF):
   The CDF, written as $F(x)$, shows the probability that the random variable $X$ is less than or equal to $x$. It's given by: $F(x) = P(X \leq x) = \sum_{t \leq x} P(X = t)$

3. Expected Value (Mean):
   The expected value, or average, of a discrete random variable $X$, denoted as $E[X]$, is calculated like this: $E[X] = \sum_{x} x \cdot P(X = x)$

4. Variance:
   Variance, written as $\text{Var}(X)$, shows how spread out the values of a random variable are. It's calculated using: $\text{Var}(X) = E[X^2] - (E[X])^2$
   Here, $E[X^2]$ is found by: $E[X^2] = \sum_{x} x^2 \cdot P(X = x)$

Continuous Probability Distributions

Continuous random variables can take any value within a certain range. Here are some important formulas for continuous distributions:

1. Probability Density Function (PDF):
   For a continuous random variable $Y$, the PDF, written as $f(y)$, is defined so that:

   • The chance that $Y$ falls between two values $a$ and $b$ is given by: $P(a < Y < b) = \int_{a}^{b} f(y) \, dy$
   • The total area under the PDF curve equals 1: $\int_{-\infty}^{\infty} f(y) \, dy = 1$

2. Cumulative Distribution Function (CDF):
   The CDF for a continuous variable is expressed as: $F(y) = P(Y \leq y) = \int_{-\infty}^{y} f(t) \, dt$

3. Expected Value (Mean):
   The expected value $E[Y]$ of a continuous random variable is calculated as: $E[Y] = \int_{-\infty}^{\infty} y \cdot f(y) \, dy$

4. Variance:
   The variance $\text{Var}(Y)$ for a continuous random variable is found using: $\text{Var}(Y) = E[Y^2] - (E[Y])^2$
   Where: $E[Y^2] = \int_{-\infty}^{\infty} y^2 \cdot f(y) \, dy$

Special Distributions

Some discrete and continuous distributions have special formulas:

• Common Discrete Distributions:

  • Binomial Distribution:
    For finding the probability of outcomes, use: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
    for $k = 0, 1, \ldots, n$.

  • Poisson Distribution:
    For certain types of events occurring, use: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$
    for $k = 0, 1, 2, \ldots$.

• Common Continuous Distributions:

  • Normal Distribution:
    The PDF is given by: $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$
    where $\mu$ is the average and $\sigma^2$ is the variance.

  • Exponential Distribution:
    The PDF is written as: $f(x; \lambda) = \lambda e^{-\lambda x}$
    for $x \geq 0$.

Understanding these key formulas is essential for solving problems related to both discrete and continuous random variables in statistics.