2.4.2 Binomial Random Variables and Its Distribution

Definition

• Binomial Random Variable: Counts the number of successes in a fixed number n of independent Bernoulli trials, each with the same probability of success p.

Conditions:

1. The trials are identical and independent.
2. Each trial results in only two possible outcomes: success or failure.
3. The probability of success p remains constant across trials.

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Probability Mass Function (PMF)

• Formula:

Where:
- is the binomial coefficient, representing the number of ways to choose k successes in n trials.
- is the probability of k successes.
- is the probability of n-k failures.

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Properties

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Examples

1. Coin Tosses:
2. A binomial experiment with multiple tosses of a coin where getting heads is considered a success.

3. If the coin is tossed n times with a success probability p for heads, then Y counts the number of heads.

4. Die Rolls:

5. Multiple rolls of a die where getting a two or less is considered a success.

6. With n rolls and a success probability , Y counts the number of times a two or less is rolled.

7. Sales Calls:

8. A salesperson makes n sales calls, with each call independently resulting in a sale (success) with probability p.
9. Y would count the total number of sales made.

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Calculation Example

• For 10 trials (n=10) and a success probability of p=0.5:
  

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