Let X B(n, p), as n → ∞, p → 0 and np → λ (0, ∞). Then show that nx p^x (1-p)^n-x n → ∞ e^-λ λ^x/x! . (10%)

5.The lifetime of an automobile battery is described by an r.v. \( X \) having normal distribution with unknown mean \( \mu \) hour and known s.d. \( \sigma=150 \) hour. Let \( X_1, X_2, \cdots, X_{25} \) be independent r.v.'s distributed as \( X \) and suppose that \( \bar{x}=1725 \) hour. Test the hypothesis \( H_0: \mu=1800 \) versus \( H_1: \mu<1800 \) at level of significance \( \alpha=0.01 \) and make a conclusion. (10%)

Let \( X \sim B(n, p) \), as \( n \to \infty \), \( p \to 0 \) and \( np \to \lambda \in (0, \infty) \). Then show that

\[ \binom{n}{x} p^x (1-p)^{n-x} \xrightarrow{n \to \infty} e^{-\lambda} \frac{\lambda^x}{x!} . \quad (10\%) \]

相關申論題

相關試卷

Let \( X \sim B(n, p) \), as \( n \to \infty \), \( p \to 0 \) and \( np \to \lambda \in (0, \infty) \). Then show that \[ \binom{n}{x} p^x (1-p)^{n-x} \xrightarrow{n \to \infty} e^{-\lambda} \frac{\lambda^x}{x!} . \quad (10\%) \]

相關申論題

相關試卷

Let \( X \sim B(n, p) \), as \( n \to \infty \), \( p \to 0 \) and \( np \to \lambda \in (0, \infty) \). Then show that

\[ \binom{n}{x} p^x (1-p)^{n-x} \xrightarrow{n \to \infty} e^{-\lambda} \frac{\lambda^x}{x!} . \quad (10\%) \]