Consider a binomial experiment with  N=20 and P=0.70.

a. Compute  F(12) (to 4 decimals).

b. Compute F(16) (to 4 decimals).

c. Compute P(X≥16) (to 4 decimals).

d. Compute  P(X ≤15)(to 4 decimals).

e. Compute E(X) .

f. Compute VAR(X) (to 1 decimal) and ?  (to 2 decimals).

Given the parameters for the binomial experiment:

- \( N = 20 \): the number of trials
- \( P = 0.70 \): the probability of success on each trial

The random variable \( X \) follows a binomial distribution \( X \sim \text{Binomial}(N=20, P=0.70) \).

### Solutions

#### Part a. Compute \( F(12) \)
\( F(12) \) is the cumulative probability \( P(X \leq 12) \).

#### Part b. Compute \( F(16) \)
\( F(16) \) is the cumulative probability \( P(X \leq 16) \).

#### Part c. Compute \( P(X \geq 16) \)
This is the probability that \( X \geq 16 \), which can be calculated as:
\[
P(X \geq 16) = 1 - F(15)
\]

#### Part d. Compute \( P(X \leq 15) \)
This is the cumulative probability \( F(15) = P(X \leq 15) \).

#### Part e. Compute \( E(X) \)
For a binomial distribution, the expected value \( E(X) \) is given by:
\[
E(X) = N \cdot P
\]
Substituting values:
\[
E(X) = 20 \cdot 0.70 = 14
\]

#### Part f. Compute \( \text{VAR}(X) \)
The variance \( \text{VAR}(X) \) of a binomial distribution is calculated as:
\[
\text{VAR}(X) = N \cdot P \cdot (1 - P)
\]
Substituting values:
\[
\text{VAR}(X) = 20 \cdot 0.70 \cdot (1 - 0.70) = 20 \cdot 0.70 \cdot 0.30 = 4.2
\]

### Calculation Approach
Let me compute the exact values for parts (a), (b), (c), and (d) using binomial cumulative probabilities.Here are the computed values:

1. **(a) \( F(12) = P(X \leq 12) \)**: 0.2277
2. **(b) \( F(16) = P(X \leq 16) \)**: 0.8929
3. **(c) \( P(X \geq 16) \)**: 0.2375
4. **(d) \( P(X \leq 15) \)**: 0.7625
5. **(e) \( E(X) \)**: 14
6. **(f) \( \text{VAR}(X) \)**: 4.2

Would you like further explanation on any part of these calculations?

---

### Follow-Up Questions
1. How would the probability values change if \( P \) were decreased?
2. What are the implications of a higher \( N \) value in this binomial experiment?
3. How would we calculate \( P(X = k) \) for a specific value of \( k \) in this context?
4. What happens to the variance if the probability \( P \) is very close to 1?
5. Why is the expected value equal to \( N \cdot P \)?

**Tip:** In a binomial distribution, as \( N \) becomes large, the distribution shape tends to approximate a normal distribution, especially when \( P \) is not too close to 0 or 1.

Consider a binomial experiment with N=20 and P=0.70.

a. Compute F(12) (to 4 decimals).

b. Compute F(16) (to 4 decimals).

c. Compute P(X≥16) (to 4 decimals).

d. Compute P(X ≤15) (to 4 decimals).

e. Compute E(X).

f. Compute VAR(X) (to 1 decimal) and ? (to 2 decimals).

Learn how to calculate cumulative probabilities F(12) and F(16), as well as P(X≥16) and P(X ≤15) for a binomial distribution with parameters N=20 and P=0.70. This tutorial also covers calculating expected value E(X) and variance VAR(X), essential for understanding binomial probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation