Unit 5: Probability - Subjective Questions

CSE333 — Combinatorial Studies-I • Practice Questions with Detailed Answers

20 questions

Define a random variable. Distinguish between discrete and continuous random variables with suitable examples.

Explain the probability mass function, probability density function, and cumulative distribution function of a random variable.

Describe the continuous uniform distribution. Derive its mean and variance for a random variable uniformly distributed over $[a,b]$ .

A bus arrives at a stop at a time uniformly distributed between 10:00 a.m. and 10:30 a.m. Find the probability that it arrives between 10:08 a.m. and 10:18 a.m. Also find the expected arrival time and standard deviation.

Explain the normal distribution and state its important properties. How is a general normal random variable converted into a standard normal variable?

The marks of students are normally distributed with mean $60$ and standard deviation $10$ . Find the probability that a randomly chosen student scores between $50$ and $75$ . Use $\Phi(1.5)=0.9332$ and $\Phi(1)=0.8413$ .

Define the exponential distribution. Derive its mean and variance, and explain its memoryless property.

The lifetime of a component follows an exponential distribution with mean $500$ hours. Find the probability that it lasts (a) more than $700$ hours and (b) an additional $300$ hours given that it has already lasted $400$ hours.

Define the Poisson distribution. Derive its mean and variance using its probability mass function.

A call center receives an average of $4$ calls every ten minutes. Assuming a Poisson distribution, find the probability of receiving (a) exactly $3$ calls and (b) at least one call in a ten-minute interval.

Define the binomial distribution and derive its mean and variance.

A fair coin is tossed $8$ times. Find the probability of obtaining (a) exactly $5$ heads and (b) at least $7$ heads. Also state the mean number of heads.

Compare the binomial and Poisson distributions. Under what conditions can the Poisson distribution approximate the binomial distribution?

Define mean, median, and mode. Explain their relationship in symmetric and moderately skewed distributions.

For the data $2,4,4,5,7,8,8,8,10$ , calculate the mean, median, mode, variance, and standard deviation. Treat the data as a population.

Explain variance and standard deviation. Derive the computational formula $\operatorname{Var}(X)=E[X^2]-(E[X])^2$ and state the effect of a linear transformation $Y=aX+b$ .

Define conditional probability and derive the multiplication rule. Distinguish between independent and mutually exclusive events.

A card is drawn from a standard deck of $52$ cards. Given that the card is a face card, find the probability that it is a king. Also find the probability that a card is both a king and a face card.

State and derive Bayes' theorem for a finite partition of the sample space.

A disease affects $1\%$ of a population. A diagnostic test has $95\%$ sensitivity and $90\%$ specificity. If a person tests positive, use Bayes' theorem to find the probability that the person actually has the disease.

Question 1

Define a random variable. Distinguish between discrete and continuous random variables with suitable examples.

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Answer 1

Random variable: A random variable is a real-valued function that assigns a numerical value to every outcome in the sample space of a random experiment.

If $S$ is the sample space, a random variable is a function $X:S\rightarrow\mathbb{R}$ .

Discrete random variable:

It assumes a finite or countably infinite set of values.
It is described by a probability mass function $P(X=x)$ .
Example: The number of heads obtained in three coin tosses can take the values $0,1,2,3$ .

Continuous random variable:

It can assume any value in an interval.
It is described by a probability density function $f(x)$ .
For any single value, $P(X=x)=0$ .
Example: The lifetime of an electric bulb may take any positive real value.

Thus, discrete variables represent countable outcomes, whereas continuous variables represent measurements over intervals.

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Question 5

Explain the normal distribution and state its important properties. How is a general normal random variable converted into a standard normal variable?

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Answer 5

A continuous random variable $X$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$ , written $X\sim N(\mu,\sigma^2)$ , if its density is

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

Important properties:

The curve is bell-shaped and symmetric about $x=\mu$ .
Mean, median, and mode are all equal to $\mu$ .
The total area under the curve is $1$ .
The tails approach the horizontal axis asymptotically.
The points of inflection are $\mu-\sigma$ and $\mu+\sigma$ .
Approximately $68.27\%$ , $95.45\%$ , and $99.73\%$ of observations lie within $1$ , $2$ , and $3$ standard deviations of the mean, respectively.

The standardization transformation is

$Z=\frac{X-\mu}{\sigma}.$

Then $Z\sim N(0,1)$ , allowing probabilities to be obtained from the standard normal table.

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Question 13

Compare the binomial and Poisson distributions. Under what conditions can the Poisson distribution approximate the binomial distribution?

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Answer 13

Binomial distribution:

Counts successes in a fixed number $n$ of independent trials.
Has parameters $n$ and $p$ .
Its support is $0,1,\ldots,n$ .
Mean: $np$ .
Variance: $np(1-p)$ .

Poisson distribution:

Counts occurrences in a fixed interval of time, space, area, or volume.
Has one parameter $\lambda$ .
Its support is $0,1,2,\ldots$ .
Mean and variance are both $\lambda$ .

The Poisson distribution approximates the binomial distribution when:

$n$ is large,
$p$ is small,
$np=\lambda$ remains finite.

Under these conditions,

$\binom{n}{x}p^x(1-p)^{n-x}\approx\frac{e^{-\lambda}\lambda^x}{x!},$

where $\lambda=np$ . A common practical guideline is $n\geq20$ and $p\leq0.05$ , although accuracy depends on the application.

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Question 14

Define mean, median, and mode. Explain their relationship in symmetric and moderately skewed distributions.

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Answer 14

Mean: The arithmetic mean is the sum of observations divided by their number. For values $x_1,x_2,\ldots,x_n$ ,

$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i.$

For a random variable, the population mean is $\mu=E[X]$ .

Median: The median is the middle value after arranging the observations in order. It divides the data into two equal parts.

Mode: The mode is the value occurring with the greatest frequency. For a continuous distribution, it is the point where the density is maximum.

Relationships:

In a symmetric unimodal distribution, $\text{mean}=\text{median}=\text{mode}$ .
In a positively skewed distribution, generally $\text{mode}<\text{median}<\text{mean}$ .
In a negatively skewed distribution, generally $\text{mean}<\text{median}<\text{mode}$ .
For a moderately skewed distribution, the empirical relation is

$\text{mode}\approx3(\text{median})-2(\text{mean}).$

The median is less affected by extreme values than the mean, while the mode is useful for identifying the most common value.

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Question 17

Define conditional probability and derive the multiplication rule. Distinguish between independent and mutually exclusive events.

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Answer 17

For events $A$ and $B$ with $P(B)>0$ , the conditional probability of $A$ given $B$ is

$P(A\mid B)=\frac{P(A\cap B)}{P(B)}.$

Rearranging gives the multiplication rule:

$P(A\cap B)=P(B)P(A\mid B).$

Similarly, if $P(A)>0$ ,

$P(A\cap B)=P(A)P(B\mid A).$

Independent events: Events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other. Thus,

$P(A\cap B)=P(A)P(B),$

or equivalently $P(A\mid B)=P(A)$ when $P(B)>0$ .

Mutually exclusive events: Events $A$ and $B$ are mutually exclusive if they cannot occur together:

$A\cap B=\varnothing\quad\text{and}\quad P(A\cap B)=0.$

Two mutually exclusive events with positive probabilities cannot be independent, because independence would require $P(A\cap B)=P(A)P(B)>0$ .

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Question 20

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Answer 20

Let $D$ denote having the disease and $T$ denote a positive test.

The given probabilities are:

$P(D)=0.01$ and $P(D^c)=0.99$ .
Sensitivity: $P(T\mid D)=0.95$ .
Specificity: $P(T^c\mid D^c)=0.90$ .
Therefore, the false-positive rate is $P(T\mid D^c)=0.10$ .

By Bayes' theorem,

$P(D\mid T)=\frac{P(D)P(T\mid D)}{P(D)P(T\mid D)+P(D^c)P(T\mid D^c)}.$

Substituting the values,

$P(D\mid T)=\frac{(0.01)(0.95)}{(0.01)(0.95)+(0.99)(0.10)}.$

$P(D\mid T)=\frac{0.0095}{0.0095+0.099}=\frac{0.0095}{0.1085}\approx0.0876.$

Thus, the probability that a person who tests positive actually has the disease is approximately $8.76\%$ . The result is relatively low because the disease is rare and false positives arise from the much larger healthy population.

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Unit4 Unit6

Unit 5: Probability - Subjective Questions

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