Probability Distributions Notes for SSC Statistics

A probability distribution shows all the possible values of a variable and the probability of each value. The binomial, Poisson and normal distributions are the three most important distributions in the SSC CGL Statistics paper. These probability distributions notes for SSC give the formulas, the mean and variance, the properties and the solved examples of each distribution in simple language.

Random Variable

• A random variable is a variable whose value depends on the result of a random experiment.
• A discrete random variable takes separate values, such as the number of heads in 5 tosses.
• A continuous random variable takes any value in a range, such as height or weight.

Summary of the Three Distributions

Binomial Distribution

• It is used when there are n independent trials, and each trial has only two outcomes: success (probability p) and failure (probability q = 1 − p).
• P(X = r) = nCr × p^r × q^{(n − r)}.
• Mean = np, variance = npq and SD = √(npq).
• The mean is always greater than the variance, because q is less than 1.

Example 1. A fair coin is tossed 5 times. Find the probability of exactly 2 heads.
P = 5C2 × (1/2)² × (1/2)³ = 10 / 32 = 5/16.

Example 2. For a binomial distribution, n = 20 and p = 0.3. Find the mean and the variance.
Mean = 20 × 0.3 = 6. Variance = 20 × 0.3 × 0.7 = 4.2.

Poisson Distribution

• It is used for rare events, such as the number of accidents on a road in a day or the number of calls in a minute.
• P(X = r) = e^−λ λ^r / r!, where e is about 2.718.
• Mean = variance = λ.
• It is the limiting case of the binomial distribution when n is large and p is very small, with np = λ.

Example 3. If λ = 2, find the probability of no event.
P(0) = e⁻² = about 0.135.

Normal Distribution

Standard Normal Distribution

• A normal variable is changed into a standard normal variable Z by Z = (X − μ) / σ.
• The standard normal distribution has mean 0 and SD 1.
• The curve was developed by Gauss, so it is also called the Gaussian distribution.

Example 4. The marks of students are normally distributed with a mean of 60 and an SD of 5. Find Z for a student who scored 70.
Z = (70 − 60) / 5 = 2.

Other Distributions

Quick Tips

• Learn the mean and variance of each distribution in one table.
• Use the binomial formula only when the trials are independent and there are only two outcomes.
• Remember that in the Poisson distribution the mean is equal to the variance.
• Remember the 68-95-99.7 rule for the normal distribution.

Quick Revision Points

• Binomial: mean = np and variance = npq.
• Poisson: mean = variance = λ.
• Normal curve is symmetric and bell-shaped, with mean = median = mode.
• Z = (X − μ) / σ.
• About 95.45 per cent of values lie within 2 SD of the mean.

Practice Questions With Answers

1. What is the mean of a binomial distribution with n = 10 and p = 0.4?
   Answer: 4
2. What is the variance of a binomial distribution with n = 10 and p = 0.4?
   Answer: 2.4
3. What is the relation between mean and variance in the Poisson distribution?
   Answer: They are equal
4. Which distribution is bell-shaped and symmetric?
   Answer: Normal distribution
5. What is the mean of the standard normal distribution?
   Answer: 0
6. What is the SD of the standard normal distribution?
   Answer: 1
7. Which distribution is used for rare events?
   Answer: Poisson distribution
8. Find Z if X = 85, mean = 70 and SD = 5.
   Answer: 3
9. What per cent of values lie within one SD of the mean in a normal distribution?
   Answer: About 68.27 per cent
10. A coin is tossed 3 times. Find P(exactly 1 head).
    Answer: 3/8

How to Practise This Chapter

Solve one problem of each distribution daily and learn the table of mean and variance. Then attempt timed sets from our practice papers and the previous year papers page. Read the previous chapter, Measures of Dispersion, and see all topics on the Statistics notes page. For the basics of probability, read the Probability notes.

Frequently Asked Questions

What is a probability distribution?

A probability distribution gives all the possible values of a random variable and the probability of each value.

What is the binomial distribution?

It is the distribution of the number of successes in n independent trials, each with the same probability of success.

What are the mean and variance of the binomial distribution?

The mean is np and the variance is npq, where q is 1 minus p.

What is the Poisson distribution?

It is the distribution of the number of rare events in a fixed time or space, such as accidents or calls.

What is the special property of the Poisson distribution?

Its mean and its variance are equal to the same value lambda.

What is the normal distribution?

It is a continuous, bell-shaped and symmetric distribution in which the mean, median and mode are equal.

What is the 68-95-99.7 rule?

In a normal distribution about 68 per cent of values lie within 1 SD, 95 per cent within 2 SD and 99.7 per cent within 3 SD of the mean.

What is the standard normal variate?

It is Z, which is equal to (X minus mean) divided by SD. It has mean 0 and SD 1.

Who developed the normal distribution?

The normal distribution is also called the Gaussian distribution after Carl Friedrich Gauss.

When does the binomial distribution become Poisson?

When n is very large and p is very small, such that np is a constant.

What is the difference between discrete and continuous variables?

A discrete variable takes separate values like 0, 1 and 2, while a continuous variable can take any value in a range.

How many questions come from probability distributions in SSC Statistics?

Usually two to four questions come from probability distributions, but the number changes from year to year.