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?

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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.