Probability distributions: A checklist for Singapore JC H2 math students

Frequently Asked Questions

What is a probability distribution in the context of H2 Mathematics?

A probability distribution describes the likelihood of different outcomes in a random experiment. It can be discrete (like the binomial or Poisson distribution) or continuous (like the normal distribution).

How do I know when to use the binomial distribution?

Use the binomial distribution when you have a fixed number of independent trials, each with only two possible outcomes (success or failure), and a constant probability of success.

What are the key properties of a Poisson distribution?

The Poisson distribution models the number of events occurring in a fixed interval of time or space, given a known average rate, and events occur independently.

When is it appropriate to approximate a binomial distribution with a normal distribution?

You can approximate a binomial distribution with a normal distribution when both *np* and *n(1-p)* are sufficiently large (usually greater than 5), where *n* is the number of trials and *p* is the probability of success.

What is the continuity correction and why is it used?

The continuity correction is used when approximating a discrete distribution (like binomial or Poisson) with a continuous distribution (like normal). It adjusts the discrete value by ±0.5 to account for the continuous nature of the normal distribution.

How do I find the mean and variance of a discrete random variable?

The mean (expected value) is calculated as Σ[*x*P(*x*)], and the variance is calculated as Σ[(*x* - μ)²P(*x*)], where *x* is the value of the random variable, P(*x*) is its probability, and μ is the mean.

What are the key differences between discrete and continuous random variables?

Discrete random variables can only take on specific, separate values (usually integers), while continuous random variables can take on any value within a given range.

How do I standardize a normal random variable?

Standardize a normal random variable *X* by subtracting the mean (μ) and dividing by the standard deviation (σ): Z = (*X* - μ) / σ. This converts *X* into a standard normal variable *Z* with a mean of 0 and a standard deviation of 1.

How can I use the Central Limit Theorem in probability problems?

The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution. This is useful for making inferences about population parameters based on sample data.