Samacheer Kalvi Class 12 Business Maths Solutions Chapter 6 Random Variable and Mathematical Expectation Ex 1.4

 

Question 1. Value which is obtained by multiplying possible values of random variable with probability of occurrence and is equal to weighted average is called
(a) Discrete value
(b) Weighted value
(c) Expected value
(d) Cumulative value
Answer: (c) Expected value
In simple words: When you multiply each possible value of a random variable by how likely it is to happen, and then add all these results together, you get the expected value. It is like finding the average outcome if you did the experiment many times.

🎯 Exam Tip: Remember that the expected value is a weighted average where probabilities are the weights, representing the long-term average outcome of a random experiment.

 

Question 2. Demand of products per day for three days are 21, 19, 22 units and their respective probabilities are 0.29, 0.40, 0.35. Profit per unit is 0.50 paisa then expected profits for three days are
(a) 21, 19, 22
(b) 21.5, 19.5, 22.5
(c) 0.29, 0.40, 0.35
(d) 3.045, 3.8, 3.85
Answer: (d) 3.045, 3.8, 3.85
Hint:

The expected value \( E(X) \) is found by summing \( x \times P(x) \) for each value.
\( E(X) = \sum_{ x } xP(x) \)
For Day 1:
\( E(X) = 21 \times 0.29 = 6.09 \)
Expected profit for Day 1 \( = 6.09 \times 0.50 = 3.045 \)
For Day 2:
\( E(X) = 19 \times 0.40 = 7.6 \)
Expected profit for Day 2 \( = 7.6 \times 0.50 = 3.8 \)
For Day 3:
\( E(X) = 22 \times 0.35 = 7.7 \)
Expected profit for Day 3 \( = 7.7 \times 0.50 = 3.85 \)
Thus, the expected profits for the three days are 3.045, 3.8, and 3.85 respectively.
In simple words: To find the expected profit for each day, first multiply the units demanded by its probability to get the expected demand. Then, multiply this expected demand by the profit per unit. This gives the average profit you can expect for each day.

🎯 Exam Tip: Remember to calculate the expected value for the quantity first, then apply the per-unit profit to find the expected profit. Be careful with decimal places in calculations.

 

Question 3. Probability which explains x is equal to or less than particular value is classified as
(a) discrete probability
(b) cumulative probability
(c) marginal probability
(d) continuous probability
Answer: (b) Cumulative Probability
In simple words: When we talk about the chance that a value \( x \) is less than or equal to a certain number, we are describing cumulative probability. It shows the total probability up to that point.

🎯 Exam Tip: Understand that cumulative probability adds up all probabilities for values up to a certain point, often written as \( P(X \le x) \).

 

Question 4. Given \( E(X) = 5 \) and \( E(Y) = -2 \), then \( E(X - Y) \) is
(a) 3
(b) 5
(c) 7
(d) None of the options
Answer: (c) 7
Hint:
The property of expectation states that \( E(X - Y) = E(X) - E(Y) \).
Using the given values:
\( E(X - Y) = 5 - (-2) \)
\( = 5 + 2 \)
\( = 7 \)
In simple words: The expected value of the difference between two random variables is simply the difference between their individual expected values. So, subtract the expected value of \( Y \) from the expected value of \( X \).

🎯 Exam Tip: Always remember the linearity property of expectation: \( E(aX \pm bY) = aE(X) \pm bE(Y) \). This makes calculating expectations of sums or differences very easy.

 

Question 5. A random variable that can assume any possible value between two points is called a
(a) discrete random variable
(b) continuous random variable
(c) discrete sample space
(d) random variable
Answer: (b) Continuous random Variable
In simple words: A continuous random variable can take on any value within a given range, even values with many decimal places. Think of measurements like height or time.

🎯 Exam Tip: The key difference between discrete and continuous random variables lies in whether they can take on all values within an interval (continuous) or only distinct, separate values (discrete).

 

Question 6. A formula or equation used to represent the probability distribution of a continuous random variable is called
(a) probability distribution
(b) distribution function
(c) probability density function
(d) mathematical expectation
Answer: (c) Probability density function
In simple words: For a continuous random variable, a special formula called a probability density function tells us how the probabilities are spread out over different values. It doesn't give a direct probability for one exact value, but for a range of values.

🎯 Exam Tip: Unlike discrete distributions, continuous distributions use probability density functions \( f(x) \) where the probability of a specific value \( P(X=x) \) is always zero. Probabilities are found over intervals using integration.

 

Question 7. If X is a discrete random variable and p(x) is the probability of X, then the expected value of this random variable is equal to
(a) \( \sum f(x) \)
(b) \( \sum[x + f(x)] \)
(c) \( \sum f(x) + x \)
(d) \( \sum xp(x) \)
Answer: (d) \( \sum xp(x) \)
In simple words: To find the expected value of a discrete random variable, you multiply each possible value by its chance of happening and then add all those products together. This sum gives you the average value you would expect over many trials.

🎯 Exam Tip: This formula, \( E(X) = \sum xp(x) \), is fundamental for discrete probability distributions and represents the mean or average of the random variable.

 

Question 8. Which of the following is not a possible value or property in a probability distribution?
(a) \( \sum p(x) > 0 \)
(b) \( \sum p(x) = 1 \)
(c) \( \sum xp(x) = 2 \)
(d) \( p(x) = -0.5 \)
Answer: (d) \( P(x) = -0.5 \)
In simple words: Probabilities must always be positive or zero. A negative probability, like -0.5, is impossible because something cannot have a less than zero chance of happening.

🎯 Exam Tip: Remember the two basic rules of probability: (1) All probabilities \( p(x) \) must be between 0 and 1 (inclusive), and (2) The sum of all probabilities \( \sum p(x) \) for all possible outcomes must equal 1.

 

Question 9. If c is a constant, then \( E(c) \) is
(a) 0
(b) 1
(c) \( c f(c) \)
(d) \( c \)
Answer: (d) \( c \)
In simple words: If you have a number that never changes (a constant), its expected value is just that number itself. There is no uncertainty about a constant value.

🎯 Exam Tip: A key property of expectation is that the expected value of a constant is the constant itself, because there's no randomness involved.

 

Question 10. A discrete probability distribution may be represented by
(a) table
(b) graph
(c) mathematical equation
(d) all of these
Answer: (d) all of these
In simple words: You can show a discrete probability distribution in different ways. You can use a table to list values and their probabilities, draw a graph to visualize it, or write a mathematical formula to describe it.

🎯 Exam Tip: Understanding that discrete probability distributions can be represented in multiple forms helps in choosing the best way to visualize or compute probabilities for different scenarios.

 

Question 11. A probability density function may be represented by:
(a) table
(b) graph
(c) mathematical equation
(d) both (b) and (c)
Answer: (d) both (b) and (c)
In simple words: A probability density function for continuous random variables can be shown using a graph, often a curve. It can also be expressed as a mathematical formula or equation that describes the curve's shape.

🎯 Exam Tip: While discrete distributions can be represented by tables, continuous distributions typically use graphs (like a curve) or mathematical equations to show how probability is distributed over a range of values.

 

Question 12. If c is a constant in a continuous probability distribution, then \( p(x = c) \) is always equal to
(a) zero
(b) one
(c) negative
(d) does not exist
Answer: (a) Zero
In simple words: For continuous probability, the chance of any single exact value happening is always zero. This is because there are infinitely many possible values. We can only find the probability that a value falls within a range.

🎯 Exam Tip: A common misconception for continuous distributions is that \( P(X=x) \) is non-zero; always remember that for a continuous random variable, the probability of any single point is zero, and probabilities are calculated over intervals.

 

Question 13. \( E[X - E(X)] \) is equal to
(a) \( E(X) \)
(b) \( V(X) \)
(c) 0
(d) None of the options
Answer: (c) 0
In simple words: The expected value of a random variable minus its own expected value is always zero. This means that, on average, the difference between a variable and its mean is zero.

🎯 Exam Tip: This is a fundamental property of expected values: the expected deviation of a random variable from its mean is always zero, \( E[X - E(X)] = E[X] - E[E(X)] = E[X] - E[X] = 0 \).

 

Question 14. \( E[X - E(X)]^2 \) is
(a) \( E(X) \)
(b) \( E(X)^2 \)
(c) \( V(X) \)
(d) \( S.D(X) \)
Answer: (c) \( V(X) \)
In simple words: This mathematical expression is the definition of variance. Variance measures how spread out the values of a random variable are from its mean.

🎯 Exam Tip: Recognize this as the definition of variance for a random variable X, which quantifies the expected squared deviation from the mean, also known as the second central moment.

 

Question 15. If the random variable takes negative values, then the negative values will have
(a) positive probabilities
(b) negative probabilities
(c) constant probabilities
(d) difficult to tell
Answer: (a) positive probabilities
In simple words: Even if a random variable has negative numbers as its possible outcomes, the probability of those negative numbers occurring must always be positive (or zero). Probabilities themselves can never be negative.

🎯 Exam Tip: Remember that probabilities are always non-negative, meaning they can be zero or any positive value up to one, regardless of whether the random variable's values are positive, negative, or zero.

 

Question 16. If we have \( f(x) = 2x \), \( 0 \le x \le 1 \), then \( f(x) \) is a
(a) probability distribution
(b) probability density function
(c) distribution function
(d) continuous random variable
Answer: (b) probability density function
In simple words: The given function, \( f(x) = 2x \) for values between 0 and 1, is a rule that describes how probabilities are spread out for a continuous random variable. It is a probability density function.

🎯 Exam Tip: For a function to be a valid probability density function, two conditions must be met: \( f(x) \ge 0 \) for all \( x \), and the integral of \( f(x) \) over its entire range must equal 1.

 

Question 17. \( \int_{ -\infty }^{\infty} f(x) dx \) is always equal to
(a) zero
(b) one
(c) \( E(X) \)
(d) \( f(x) + 1 \)
Answer: (b) one
In simple words: When you add up (integrate) all the probabilities from a probability density function over its entire possible range, the total must always be 1. This means there is a 100% chance of *some* outcome happening.

🎯 Exam Tip: This is a fundamental property of all probability density functions (PDFs) for continuous random variables: the area under the entire curve must always sum to 1.

 

Question 18. A listing of all the outcomes of an experiment and the probability associated with each outcome is called
(a) probability distribution
(b) probability density function
(c) attributes
(d) distribution function
Answer: (a) Probability distribution
In simple words: A probability distribution is like a map that shows all the possible results of an experiment and how likely each result is. It helps us understand the chances of different things happening.

🎯 Exam Tip: Clearly distinguish between a probability distribution (the overall mapping of outcomes to probabilities) and a probability density function (a specific formula for continuous distributions).

 

Question 19. Which one is not an example of random experiment?
(a) A coin is tossed and the outcome is either a head or a tail
(b) A six-sided die is rolled
(c) Some number of persons will be admitted to a hospital emergency room during any hour.
(d) All medical insurance claims received by a company in a given year.
Answer: (d) All medical insurance claims received by a company in a given year.
In simple words: A random experiment is one where you don't know the exact outcome beforehand, but you know all possible outcomes. Counting *all* claims received in a year is a fixed number, not something with uncertain individual outcomes, making it not a random experiment.

🎯 Exam Tip: A random experiment must have outcomes that cannot be predicted with certainty before the experiment is performed, even if all possible outcomes are known.

 

Question 20. A set of numerical values assigned to a sample space is called
(a) random sample
(b) random variable
(c) random numbers
(d) random experiment
Answer: (b) random variable
In simple words: A random variable is a way to turn the results of a random experiment (like rolling a die) into numbers. For example, if you flip a coin, heads could be 1 and tails could be 0.

🎯 Exam Tip: A random variable essentially assigns a numerical value to each outcome in a sample space, making it easier to perform mathematical analysis on random phenomena.

 

Question 21. A variable which can assume finite or countably infinite number of values is known as
(a) continuous
(b) discrete
(c) qualitative
(d) none of them
Answer: (b) Discrete
In simple words: A discrete variable can only take certain, distinct values, like whole numbers. It cannot take any value in between. For example, the number of children in a family is a discrete variable.

🎯 Exam Tip: Discrete random variables are typically associated with counting, where values are integers and can be listed, while continuous random variables are associated with measuring, where values can be any real number within an interval.

 

Question 22. The probability function of a random variable is defined as

Then k is equal to
(a) zero
(b) \( \frac { 1 }{4} \)
(c) \( \frac { 1 }{15} \)
(d) one
Answer: (c) \( \frac { 1 }{15} \)
Hint:
For any valid probability distribution, the sum of all probabilities must be equal to 1.
\( \sum P(x_i) = 1 \)
So, we add up all the given probabilities:
\( P(x = -1) + P(x = -2) + P(x = 0) + P(x = 1) + P(x = 2) = 1 \)
\( k + 2k + 3k + 4k + 5k = 1 \)
Now, sum all the terms with \( k \):
\( 15k = 1 \)
To find \( k \), divide both sides by 15:
\( k = \frac { 1 }{15} \)
In simple words: The basic rule for any probability list is that all the probabilities must add up to exactly 1. We added all the 'k' terms together, set the total equal to 1, and then solved for 'k'.

🎯 Exam Tip: Always remember the fundamental rule that the sum of all probabilities in any discrete probability distribution must equal 1. This is a crucial step for solving problems involving unknown constants like 'k'.

 

Question 23. If \( P(x) = \frac{1}{10} \) and \( x = 10 \), then \( E(X) \) is
(a) zero
(b) \( \frac { 6 }{8} \)
(c) 1
(d) -1
Answer: (c) 1
Hint:
The expected value \( E(X) \) for a single outcome \( x \) with probability \( P(x) \) is given by \( x \times P(x) \).
Given \( P(x) = \frac{1}{10} \) and \( x = 10 \).
So, \( E(X) = 10 \times \frac{1}{10} \)
\( = 1 \)
In simple words: To find the expected value, you simply multiply the value of \( x \) by its probability \( P(x) \). In this case, multiplying 10 by one-tenth gives us 1.

🎯 Exam Tip: For a single event with a specific value and its probability, the expected value is simply their product. For multiple events, it's the sum of such products.

 

Question 24. A discrete probability function \( p(x) \) is always
(a) non-negative
(b) negative
(c) one
(d) zero
Answer: (a) non-negative
In simple words: A probability function must always give results that are zero or positive. It can never give a negative number, because probabilities cannot be less than nothing.

🎯 Exam Tip: Remember the fundamental rules of probability: (1) \( 0 \le p(x) \le 1 \) for all outcomes \( x \), and (2) \( \sum p(x) = 1 \). Non-negative means greater than or equal to zero.

 

Question 25. In a discrete probability distribution the sum of all the probabilities is always equal to
(a) zero
(b) one
(c) minimum
(d) maximum
Answer: (b) one
In simple words: If you add up the chances of all possible outcomes in a discrete probability distribution, the total will always be 1. This means there is a 100% chance that one of the possible outcomes will happen.

🎯 Exam Tip: This is a core axiom of probability theory: the sum of probabilities for all exhaustive and mutually exclusive outcomes in a sample space must equal 1.

 

Question 26. An expected value of a random variable is equal to its
(a) variance
(b) standard deviation
(c) mean
(d) con variance
Answer: (c) mean
In simple words: The expected value of a random variable is basically the average value you would get if you repeated the random experiment many times. So, "expected value" is just another name for the mean.

🎯 Exam Tip: The terms "expected value" and "mean" (often denoted as \( \mu \)) are used interchangeably in probability and statistics; they represent the central tendency of a random variable.

 

Question 27. A discrete probability function \( p(x) \) is always non-negative and always lies between
(a) 0 and \( \infty \)
(b) 0 and 1
(c) -1 and +1
(d) \( -\infty \) and \( \infty \)
Answer: (b) 0 and 1
In simple words: Any probability, whether for a discrete or continuous event, must always be a number between 0 and 1. A probability of 0 means it cannot happen, and 1 means it will definitely happen.

🎯 Exam Tip: Always remember the range for valid probability values; they can never be negative and can never exceed one, reflecting the certainty of an event.

 

Question 28. The probability density function \( p(x) \) cannot exceed
(a) zero
(b) one
(c) mean
(d) infinity
Answer: (b) One
In simple words: A probability density function value can technically be greater than 1 at some points for continuous variables, but the *integral* (total area) over the entire range must be 1. The maximum *probability* for any event cannot exceed 1. This question is likely implying probabilities, not the function value itself.

🎯 Exam Tip: While a probability density function \( f(x) \) can sometimes have values greater than 1 (especially for narrow ranges), the *probability* of any event (an interval) derived from it must always be between 0 and 1. The sum of all probabilities is 1.

 

Question 29. A variable representing measurements (like height or temperature) in a country is a random variable of the type
(a) discrete random variable
(b) continuous random variable
(c) both (a) and (b)
(d) neither (a) not (b)
Answer: (b) Continuous random variable
In simple words: Variables like height or temperature can take on any value within a range, not just specific, separate numbers. This makes them continuous random variables.

🎯 Exam Tip: Remember that continuous variables are typically associated with measurements, where values can be infinitely divisible, whereas discrete variables are associated with counting, having distinct, separate values.

 

Question 30. The distribution function \( F(x) \) is equal to
(a) \( P(X = x) \)
(b) \( P(X \le x) \)
(c) \( P (X \ge x) \)
(d) all of these
Answer: (b) \( P(X \le x) \)
In simple words: The distribution function, also called the cumulative distribution function, tells us the probability that a random variable \( X \) will be less than or equal to a certain value \( x \). It accumulates probabilities up to that point.

🎯 Exam Tip: The cumulative distribution function (CDF), \( F(x) = P(X \le x) \), is a fundamental concept used to find probabilities over intervals for both discrete and continuous random variables.