Theoretical Distributions - ICAI Foundation Statistics

Comprehensive practice flashcards covering theoretical probability distributions (Binomial, Poisson, and Normal distributions) based on the ICAI Foundation Statistics exercises.

What is true regarding the existence of a theoretical probability distribution?

It exists in theory and exists in real life (both in theory and in real life).

What two general forms can a probability distribution take?

It may be discrete or continuous.

Which distribution is an important example of a discrete probability distribution?

Poisson distribution (as well as Binomial distribution).

Which distribution is considered the most important continuous probability distribution?

Normal distribution.

In statistical theory, what is a parameter a characteristic of?

A population.

What is an example of a parameter?

Population mean.

In probability theory, how is a trial defined?

An attempt to produce an outcome which is neither certain nor impossible.

What are the two defining characteristics of Bernoulli trials?

Each trial is associated with just two possible outcomes, and the trials are independent.

What is the probability mass function of a binomial distribution?

$$f(x) = {^n C_x} p^x q^{n-x}$$

If $x$ is a binomial variable with parameters $n$ and $p$, what values can $x$ assume?

Any whole number between $0$ and $n$, both inclusive.

Under what condition is a binomial distribution symmetrical?

When $p = 0.5$.

What is the mean of a binomial distribution with parameters $n$ and $p$?

$$np$$

What is the formula for the variance of a binomial distribution with parameters $n$ and $p$?

$np(1 - p)$ (or $npq$)

What is an example of a bi-parametric discrete probability distribution?

Binomial distribution (having parameters $n$ and $p$).

Under what condition are the mean and mode of a binomial distribution equal?

When $q = 0.50$ (or $p = 0.50$).

How does the mean of a binomial distribution compare to its variance?

The mean is always more than its variance.

How many modes can a binomial distribution have?

One mode or two modes (it may be unimodal or bimodal).

What is the maximum value of the variance of a binomial distribution with parameters $n$ and $p$?

$\frac{n}{4}$ (which occurs when $p = q = 0.5$).

Which method is usually applied for fitting a binomial distribution?

Method of moments.

Why is the statement 'the probability of having success in a small time interval $(t, t + dt)$ is $kt$ for a positive constant $k$' not a valid condition of the Poisson model?

In a Poisson process, the probability of success in a small time interval is proportional to the interval length ($k \, dt$), not $kt$.

Which probability distribution is uniparametric?

Poisson distribution (having only one parameter, $m$).

What is the relationship between the mean and variance for a Poisson distribution?

The mean and variance are equal.

What modality can a Poisson distribution exhibit?

It may be unimodal or bimodal.

What is the skewness characteristic of a Poisson distribution?

It is always positively skewed.

Under what conditions can a binomial distribution with parameters $n$ and $p$ be approximated by a Poisson distribution?

When $n \rightarrow \infty$ and $p \rightarrow 0$ so that $np$ remains finite.

How is a Poisson distribution fitted to an observed frequency distribution?

By equating the Poisson parameter to the mean of the frequency distribution.

What is the probability density function of a normal random variable $x$ with mean $\mu$ and standard deviation $\sigma$?

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$ for $-\infty < x < \infty$.

What is the total area under the normal curve?

One ($1$).

What is the shape of the normal curve?

Bell-shaped.

What is the skewness of the normal curve?

It is symmetrical (skewness is zero).

What are the areas of the normal curve from $-\infty$ to $\mu$ and from $\mu$ to $\infty$?

The area between $-\infty$ to $\mu$ is $0.50$, and the area between $\mu$ to $\infty$ is $0.50$.

How is the cumulative distribution function $F(x)$ of a random variable $X$ defined?

$$F(x) = P(X \le x)$$

How do the mean and mode of a normal distribution relate to each other?

They are always equal (mean = median = mode).

What is the mean deviation about the median of a standard normal variate?

$0.80$ (since $\text{MD} \approx 0.80\sigma$ and $\sigma = 1$ for a standard normal variate).

What is the quartile deviation of a normal distribution with mean $10$ and standard deviation $4$?

$2.70$ (calculated as $\text{QD} \approx 0.675 \times \sigma = 0.675 \times 4 = 2.70$).

What are the points of inflexion for a standard normal distribution?

$-1$ and $1$.

In standard normal distribution notation, what area does the symbol $\Phi(a)$ indicate?

The area of the standard normal curve between $-\infty$ to $a$.

What portion of the total area of a normal distribution is covered by the interval $(\mu - 3\sigma, \mu + 3\sigma)$?

All but $0.27\%$ area of a normal distribution (i.e., $99.73\%$).

Which probability distribution is typically followed by the number of misprints per page of a thick book?

Poisson distribution.

Which distribution models the results of ODI matches between two teams like India and Pakistan?

Binomial distribution.

Which distribution do the wages of workers of a factory typically follow?

Normal distribution.

If $X$ and $Y$ are two independent normal random variables, what is the distribution of $(X + Y)$?

Normal distribution.