Probability Distributions: One Mark Questions, Book Back Answers and Notes

About Probability Distributions

Medium ~180 min study

In Class 11 you listed sample spaces and counted favourable outcomes. That works for a coin or a die, but it gets clumsy fast. A random variable fixes this. It is a rule that gives every outcome a real number, so "toss three coins" becomes "X = number of heads", and X can take the values 0, 1, 2 or 3. Once outcomes are numbers, you can add them, square them and integrate over them.

The unit splits into two tracks. A discrete random variable takes values you can count, like the number of defective bulbs in a box. A continuous random variable takes any value in an interval, like the waiting time at a bus stop. For discrete variables you work with sums; for continuous ones you work with integrals. Both tracks lead to the same ideas: the distribution function, the mean and the variance.

Book back MCQs in this unit are mostly short calculations. You may be asked to find an unknown constant \(k\) in a mass or density function, read a probability off a distribution function, or find the mean and variance of a binomial variable using \(np\) and \(np(1-p)\). Before you pick an option, check that your probabilities add up to 1; that single check catches many wrong answers.

What you'll learn

• Define a random variable and tell whether it is discrete or continuous.
• Write down the probability mass function or density function for a given experiment and find any unknown constant.
• Find the cumulative distribution function from a mass or density function, and go back the other way.
• Calculate the mean and variance of a random variable using sums or integrals.
• Use the Bernoulli and binomial distributions to find probabilities, means and variances.
• Explain the mean as the long-run average value over many repetitions of an experiment.

Before you start

• Sample spaces, events and basic probability rules from Class 11.
• Set notation and the idea of a function.
• Definite integration, for handling continuous random variables.

Probability Distributions explained

What This Unit Covers

Random Variables

A random variable is a function from the sample space to the set of real numbers. Toss a coin twice and let X be the number of heads. The outcome HH goes to 2, HT and TH go to 1, and TT goes to 0. Nothing about the experiment changes; we have only attached numbers to the outcomes. This is what lets us use algebra and calculus on questions about chance. Keep in mind that X is a function, not a single number, and different outcomes can map to the same value.

Discrete Random Variables and the Probability Mass Function

A discrete random variable takes a finite or countably infinite set of values. Its probability mass function gives \(P(X = x)\) for each value. Two conditions must hold: every probability is non-negative, and all of them add up to 1. Many MCQs give a table with an unknown \(k\) and expect you to use the second condition to find it. The cumulative distribution function is \(F(x) = P(X \le x)\), a running total of the mass function. To go back from \(F\) to the mass function, subtract consecutive values: \(P(X = x_i) = F(x_i) - F(x_{i-1})\).

Continuous Random Variables and the Probability Density Function

A continuous random variable can take any value in an interval. Here we use a probability density function \(f(x)\), which is non-negative and has total integral 1 over the real line. The probability that X lies between \(a\) and \(b\) is the area under \(f(x)\) from \(a\) to \(b\). The probability of X taking one exact value is zero, so \(P(a \le X \le b)\) and \(P(a < X < b)\) are equal. The distribution function is \(F(x) = \int_{-\infty}^{x} f(t)\,dt\), and wherever \(F\) is differentiable, \(F'(x) = f(x)\). Time, length and weight are the usual examples.

Mathematical Expectation (Mean)

The mean or expected value \(E(X)\) is the average value of X you would get if the experiment were repeated a very large number of times. For a discrete variable, multiply each value by its probability and add: \(E(X) = \sum x\,p(x)\). For a continuous variable, integrate: \(E(X) = \int x\,f(x)\,dx\). The mean need not be a value X can actually take. For a fair die, \(E(X) = 3.5\), yet no throw shows 3.5. Two properties save time: \(E(aX + b) = aE(X) + b\) and \(E(c) = c\) for a constant.

Variance and Standard Deviation

Variance measures how widely the values of X are spread around the mean. It is defined as \(V(X) = E[(X - \mu)^2]\), and in practice you compute it as \(V(X) = E(X^2) - [E(X)]^2\). The standard deviation is the positive square root of the variance and has the same units as X. Remember \(V(aX + b) = a^2 V(X)\): adding a constant shifts the values but does not change the spread. Students often write \(aV(X)\) here and lose the mark.

Bernoulli and Binomial Distributions

A Bernoulli trial has exactly two outcomes, success with probability \(p\) and failure with probability \(q = 1 - p\). The Bernoulli random variable takes the value 1 for success and 0 for failure. Its mean is \(p\) and its variance is \(pq\). Repeat the trial \(n\) times independently with the same \(p\), and let X count the successes. Then X follows the binomial distribution with parameters \(n\) and \(p\): \(P(X = x) = \binom{n}{x} p^x q^{n-x}\) for \(x = 0, 1, \dots, n\). Its mean is \(np\) and its variance is \(npq\). Since \(0 < p < 1\) gives \(q < 1\), the variance of a binomial variable is less than its mean, which is a quick way to reject impossible options in an MCQ.

Common mistakes to avoid

• Giving a non-zero answer for P(X = a) when X is continuous. For a continuous variable the probability at a single point is always zero.
• Forgetting to use the condition that the probabilities add to 1, or that the density integrates to 1, when finding an unknown constant k.
• Reading F(x) as P(X = x). F(x) is a running total; for a discrete variable subtract F at the previous value to get the point probability.
• Adding values of a density function as if it were a mass function, instead of integrating over the interval.
• Writing V(aX + b) = aV(X) + b. The correct result is a²V(X), because a constant shift does not change the spread.
• Using the binomial formula when trials are not independent or the success probability changes from trial to trial, as in drawing cards without replacement.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

What is the difference between discrete and continuous random variables?

A discrete random variable takes values you can list and count, like the number of heads in five tosses. A continuous random variable can take any value in an interval, like height or time. For discrete variables you add probabilities; for continuous variables you integrate a density function.

How do I calculate the mean of a random variable?

For a discrete variable, multiply each value by its probability and add the results. For a fair die this gives (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5. For a continuous variable, integrate x times the density function over the whole range of X.

What does the cumulative distribution function tell us?

F(x) gives the probability that X is less than or equal to x. It never decreases, it tends to 0 on the far left and to 1 on the far right. It is handy for interval questions, since the probability that X lies in the interval (a, b] is F(b) − F(a).

Why is the probability of a single point zero in continuous distributions?

Probability for a continuous variable is the area under the density curve. Over a single point the width of the region is zero, so the area, and therefore the probability, is zero. That is why we only ask for probabilities over intervals.

What conditions are needed for a binomial distribution?

You need a fixed number of trials n, each trial has only two outcomes (success or failure), the trials are independent, and the probability of success p is the same in every trial. When all four hold, the number of successes follows the binomial distribution.

How are variance and standard deviation related?

The variance is the expected value of the squared distance from the mean. The standard deviation is its positive square root. We often quote the standard deviation because it is in the same units as X, while the variance is in squared units.