Question

Consider a binomial experiment with n = 10 and p = 0.20.

(a) Compute f(0). (Round your answer to four decimal places )
f(0) = ?
(b) Compute f(2). (Round your answer to four decimal places.)
f(2) = ?
(c) Compute P(x ≤ 2). (Round your answer to four decimal places)
P(x ≤ 2) = ?
(d) Compute P(x ≤ 1). (Round your answer to four decimal places)
P(x ≤ 1) = ?
(e) Compute E(x)
E(x) = ?
(f) Compute Var(x) and σ. (Round your answer to four decimal places)
Var(x) = ?
σ = ?

Answer 1

For a binomial experiment with n = 10 and p = 0.20, f(0) = 0.1074, f(2) = 0.4719, P(x ≤ 2) = 0.8480, P(x ≤ 1) = 0.3761, E(x) = 2, Var(x) = 1.60, and σ = 1.26.

Step-by-step explanation:

In a binomial experiment with n = 10 and p = 0.20:

a) To compute f(0), we use the formula f(x) = nCx * p^x * q^(n-x). Here, f(0) = 10C0 * (0.20)^0 * (0.80)^(10-0) = 1 * 1 * 0.1074 = 0.1074.

b) To compute f(2), we use the same formula. Here, f(2) = 10C2 * (0.20)^2 * (0.80)^(10-2) = 45 * 0.04 * 0.2621 = 0.4719.

c) To compute P(x ≤ 2), we add the probabilities of f(0), f(1), and f(2). P(x ≤ 2) = f(0) + f(1) + f(2) = 0.1074 + 0.2687 + 0.4719 = 0.8480.

d) To compute P(x ≤ 1), we add the probabilities of f(0) and f(1). P(x ≤ 1) = f(0) + f(1) = 0.1074 + 0.2687 = 0.3761.

e) The expected value E(x) can be calculated using the formula E(x) = n * p. Here, E(x) = 10 * 0.20 = 2.

f) The variance Var(x) can be calculated using the formula Var(x) = n * p * q and the standard deviation σ can be calculated as σ = √Var(x). Here, Var(x) = 10 * 0.20 * 0.80 = 1.60 and σ = √1.60 = 1.26.