Analysis of probability models:

• Incorrect: Uniform distribution assigns equal probability across all outcomes without binomial compounding.
• Incorrect: Exponential distribution is a continuous distribution modeling time between Poisson events.
• Incorrect: Poisson distribution models rare events over continuous time/space where $n$ is very large and $p$ very small.
• Correct: The scenario satisfies all criteria of a Binomial experiment $B(n=30, p=0.25)$:
• Fixed number of identical trials ($n = 30$).
• Each question trial has only two mutually exclusive outcomes: success (correct guess) or failure (incorrect guess).
• Constant probability of success on each trial ($p = 1/4 = 0.25$).
• Independent guesses between questions.

Relationship between regression slopes and correlation coefficient:

• $1/4$ is the reciprocal slope of the equation, not necessarily the correlation coefficient.
• Correlation coefficient $r$ cannot exceed $+1$, so it cannot be $4$.
• In the regression of $X$ on $Y$, $b_{xy} = +4$. A fundamental statistical theorem states that the regression coefficients $b_{yx}$, $b_{xy}$, and the Pearson correlation coefficient $r$ must all share the exact same mathematical sign. Since $b_{xy} \gt 0$, $r$ must be strictly positive ($r \gt 0$).
• $r$ cannot exceed $1$ by definition ($-1 \le r \le +1$).

Mathematical boundaries of Pearson's $r$:

• By Cauchy-Schwarz inequality, the Pearson product-moment correlation coefficient is strictly bounded such that $-1 \le r \le +1$. A value of $+1$ indicates perfect positive linear correlation, $-1$ indicates perfect negative linear correlation, and $0$ indicates absence of linear correlation.
• Covers only negative correlation.
• Covers only positive correlation.
• $-\infty$ to $+\infty$ is the range of regression coefficients or covariance, not correlation.

Central measures in standard normal distribution:

• $1$ is the standard deviation (and variance) of the standard normal distribution.
• The median is precisely zero, not any arbitrary positive number.
• $5$ is incorrect.
• The standard normal distribution $Z \sim N(0, 1)$ is perfectly symmetric about $0$. In any symmetric unimodal distribution, mean $=$ median $=$ mode. Since the mean is $\mu = 0$, the median is identically 0.