# PROPERTIES OF SOME PROBABILITY DISTRIBUTIONS

## PROPERTIES OF THE NORMAL DISTRIBUTION

1. A normal distribution is a continuous probability distribution for a random variable.

2. The graph of a normal distribution is called the normal curve, which has all of the following properties:

1. The mean, median and mode are equal.
2. The normal curve is bell-shaped and is symmetric about the mean.

c. The total area under the curve is equal to one.

d. The normal curve approaches, but never touches, the x-axis.

e.The standard normal distribution is a normal distribution with a mean of 0, standard deviation of 1.

## PROPERTIES OF THE POISSON DISTRIBUTION

1. It is named after French mathematician Siméon Denis Poisson.
2. It is a discrete probability distribution.
3. The Poisson distribution is popular for modelling the number of times an event occurs in an interval of time or space.
4. Given the mean number of successes (λ) that occur in a specified region, we can compute the Poisson probability for the random variable X based on the following formula:  $\centering P(X=r) = \frac{e^{-\lambda} \times \lambda^r}{r!}$
5. The mean of the distribution is equal to λ.
6. The variance is also equal to λ.

## PROPERTIES OF THE BINOMIAL DISTRIBUTION

1. It is a discrete probability distribution.
2. A Bernoulli trial is an experiment that has exactly two outcomes. For example, the experiment of tossing a coin has two outcomes viz head or tail. One of these outcomes is called success and the other is defined as failure.
3. The Binomial distribution is used to model the probability of an experiment where a Bernoulli trial is repeated multiple times. The random variable is the number of successes when a Bernoulli trial is repeated ‘n’ times.
4. Given that a Bernoulli trial is repeated ‘n’ times with probability of success being ‘p’ and probability of failure being ‘q’, we can compute the Binomial probability for the random variable X based on the following formula:  $\displaystyle P(X=r) = \binom{n}{r} \times p^r \times q^{n-r}$ where ‘r’ is the number of successes.
5. The mean of the distribution is equal to $np$.
6. The variance is also equal to $npq$.
7. For p and q as defined above, $p+q =1$