Sampling, data types, bias and outliers：練習問題

A set of data has a mean of \( 15 \) and a standard deviation of \( 4 \). If every value in the data set is multiplied by \( 2 \) and then increased by \( 3 \), what are the new mean and the new standard deviation?

Two events \( A \) and \( B \) are independent such that \( P(A) = 0.4 \) and \( P(A \cup B) = 0.7 \). Find \( P(B) \).

The discrete random variable \(X\) follows a binomial distribution \(B(n, 0.4)\). Given that the probability of obtaining no successes is \(P(X = 0) = 0.1296\), find the value of \(n\).

Two events \( A \) and \( B \) are such that \( P(A) = 0.35 \) and \( P(B) = 0.45 \). If \( A \) and \( B \) are mutually exclusive, find \( P(A \cup B) \).

A discrete random variable \(X\) has an expected value \(E(X) = 2\) and \(E(X^2) = 4.5\). Find the variance of the random variable \(Y = 2X - 3\).

The discrete random variable \( X \) has a probability distribution given by \( P(X=1) = 0.15 \), \( P(X=2) = 0.45 \), and \( P(X=3) = k \). Find the value of the constant \( k \).

Two events \( A \) and \( B \) are defined such that \( P(A) = 0.6 \), \( P(B) = 0.5 \), and \( P(A' \cap B') = 0.1 \). Calculate the value of \( P(A \cap B) \).

A group of 10 observations has a mean of 5 and a variance of 2. A second group of 15 observations has a mean of 10 and a variance of 5. Calculate the combined variance of all 25 observations.

The weights of apples in an orchard are normally distributed with mean \( μ \) and standard deviation \( σ \). It is observed that 10% of the apples weigh more than 200g, and 20% of the apples weigh less than 140g.

a) Set up two simultaneous equations for \( μ \) and \( σ \) using z-scores.
b) Find the value of \( μ \) and \( σ \).
c) An apple is chosen at random. Find the probability that it weighs between 150g and 180g.

A continuous random variable \( X \) has a probability density function given by:
\( f(x) = \begin{cases} k \cos^2(x) & 0 \le x \le \pi \\ 0 & \text{otherwise} \end{cases} \)
(a) Show that the value of the constant \( k \) is \( \frac{2}{\pi} \). (3 points)
(b) Find the cumulative distribution function \( F(x) \) for \( 0 \le x \le \pi \). (2 points)
(c) Calculate the probability \( P\left( \frac{\pi}{4} \le X \le \frac{3\pi}{4} \right) \). (2 points)