5.1 Random Variable
Discrete and Continuous Random Variables
- Discrete Random Variable: A variable with countable outcomes (e.g., number of heads obtained, number of defects).
- Continuous Random Variable: A variable that takes any real continuous value within an interval (e.g., height, weight, time).
Probability Distribution of a Discrete Random Variable
For a discrete random variable $X$ taking values $x_1, x_2, \dots, x_n$:
$$P(X = x_i) \ge 0 \quad \text{for all } i$$
$$\sum P(X = x_i) = 1$$
5.2 Binomial Distribution
Conditions for a Binomial Experiment
- The experiment consists of $n$ identical and independent trials.
- Each trial results in only two possible outcomes: Success ($S$) or Failure ($F$).
- The probability of success, $p$, remains constant for each trial.
- Probability of failure is $q = 1 - p$.
Binomial Probability Formula
If $X \sim B(n, p)$ represents the number of successes in $n$ trials:
$$P(X = r) = \, ^n C_r \, p^r \, q^{n-r}, \quad r = 0, 1, 2, \dots, n$$
Mean, Variance, and Standard Deviation of Binomial Distribution
- Mean ($\mu$ or $E(X)$): $\mu = np$
- Variance ($\sigma^2$): $\sigma^2 = npq$
- Standard Deviation ($\sigma$): $\sigma = \sqrt{npq}$
5.3 Normal Distribution
Properties of Standard Normal Distribution $N(0, 1)$
- A continuous distribution that is symmetrical and bell-shaped about the mean $\mu$.
- Mean = Median = Mode = $\mu$.
- Total area under the normal curve is equal to $1$.
Standard Normal Distribution Variable ($Z$)
Any continuous normal random variable $X \sim N(\mu, \sigma^2)$ is standardized to $Z \sim N(0, 1)$ using:
$$Z = \frac{X - \mu}{\sigma}$$
Finding Probabilities from Z-Score
- $P(Z > k)$ or $P(Z < -k)$ can be read directly from the standard normal cumulative distribution tables or computed using $1 - P(Z \le k)$.
- $P(a < Z < b) = P(Z < b) - P(Z < a)$.
- Symmetry rule: $P(Z < -k) = P(Z > k)$.