وَأَحْصَىٰ كُلَّ شَيْءٍ عَدَدًا
"And He has enumerated everything in numbers." (Allah perfectly counts and knows the exact measure of all creations).
Q1. Pseudo random techniques are used to:
A. Generate random
numbers ✓ Correct Answer
B. Find central tendency
C. Find dispersion
D. None
Explanation: Pseudo random number generation is a computational
technique specifically designed to generate sequences of random-looking
numbers.
Q2. Pseudo random digits are:
A. Truly random
B. Deterministic
C. Non-random
D. Both (b) and (c) ✓ Correct Answer
Explanation: Pseudo random digits are generated by a deterministic
algorithm/formula, so despite appearing random, they are actually non-random
and deterministic.
Q3. Each random digit has:
A. The same probability ✓ Correct Answer
B. Different chances
C. Zero probability
D. Both (a) and (c)
Explanation: In a properly generated random digit sequence, every
digit (0-9) has an equal chance of appearing.
Q4. The probability of each random digit (0, ..., 9)
is:
A. 0.1 ✓ Correct Answer
B. 0.9
C. 0.8
D. 0.7
Explanation: With 10 equally likely digits (0-9), each has a
probability of 1/10 = 0.1.
Q5. Random numbers are used for the selection of a:
A. Random sample ✓ Correct Answer
B. Non-random sample
C. Judgment sample
D. None
Explanation: Random numbers are the standard tool for selecting a
random sample from a population.
Q6. The distribution of random numbers is:
A. Uniform ✓ Correct Answer
B. Binomial
C. Hypergeometric
D. None
Explanation: Since every digit has an equal chance of occurring,
random numbers follow a uniform distribution.
Q7. The probability distribution of the numbers
appearing when a die is thrown is:
A. Uniform ✓ Correct Answer
B. Normal
C. Hypergeometric
D. None
Explanation: Each face of a fair die (1 through 6) has an equal
probability of 1/6, giving a uniform distribution.
Q8. Every random experiment has:
A. Only one outcome
B. Two or more outcomes ✓ Correct Answer
C. No outcome
D. None
Explanation: A random experiment, by definition, must have more than
one possible outcome — otherwise there'd be no uncertainty.
Q9. A variable whose value depends upon the outcomes
of a random experiment is called a:
A. Stochastic variable
B. Random variable
C. Chance variable
D. All of these ✓ Correct Answer
Explanation: Stochastic variable, random variable, and chance variable
are all different names for the same concept.
Q10. A random variable may assume:
A. Negative values
B. Positive values
C. Both (a) and (b) ✓ Correct Answer
D. None
Explanation: A random variable can take on both negative and positive
values, depending on the underlying experiment.
Q11. A random variable is also called a:
A. Stochastic variable
B. Chance variable
C. Both (a) and (b) ✓ Correct Answer
D. None
Explanation: Both 'stochastic variable' and 'chance variable' are
alternate names for a random variable.
Q12. A random variable assuming only a finite number
of values is called a:
A. Continuous random variable
B. Discrete random
variable ✓ Correct Answer
C. Discontinuous variable
D. None
Explanation: A variable that can only take a countable, finite set of
values is classified as a discrete random variable.
Q13. The lifetime of a light bulb is an example of a:
A. Continuous random
variable ✓ Correct Answer
B. Discrete random variable
C. Discontinuous variable
D. Both (b) and (c)
Explanation: Lifetime can take any value in a range (e.g. 100.5 hours,
200.37 hours), making it a continuous random variable.
Q14. A discrete random variable takes only values
that are:
A. Countable ✓ Correct Answer
B. Infinite
C. Uncountable
D. None
Explanation: By definition, discrete random variables take on a
countable set of distinct values.
Q15. The probability distribution of a discrete
random variable is called its:
A. Continuous (probability
density) function
B. Discrete (probability
mass) function ✓ Correct Answer
C. Both (a) and (b)
D. None
Explanation: For discrete random variables, the distribution is
described by the probability mass function (p.m.f.).
Q16. The probability function of a random variable is
non-negative:
A. Always true ✓ Correct Answer
B. Partially true
C. False
D. None
Explanation: Probabilities can never be negative, so a valid
probability function is always non-negative.
Q17. For a random variable X, Σp(x) =
A. Always zero
B. Always 1 ✓ Correct Answer
C. Greater than 1
D. Less than 1
Explanation: The sum of probabilities over all possible values of a
random variable must always equal 1.
Q18. A probability function can never be:
A. Negative ✓ Correct Answer
B. Positive
C. Undefined
D. None
Explanation: Since probabilities lie between 0 and 1, a valid
probability function can never take a negative value.
Q19. Probability is obtained by inserting the value
of the random variable directly into the probability function in the case of a:
A. Discrete variable ✓ Correct Answer
B. Continuous variable
C. Discontinuous variable
D. None
Explanation: For discrete variables, you substitute the specific value
into the probability mass function to get its probability.
Q20. A continuous random variable almost always arises
in connection with:
A. Counting
B. Measuring
C. Weighing
D. Both (b) and (c) ✓ Correct Answer
Explanation: Continuous variables typically come from measurement
processes like measuring length, weight, time, etc.
Q21. The probability distribution of a continuous
random variable is called its:
A. Probability density
function ✓ Correct Answer
B. Probability mass function
C. Both (a) and (b)
D. None
Explanation: Continuous random variables are described using a
probability density function (p.d.f.), unlike discrete variables which use a
mass function.
Q22. The total area under the (probability) density
curve is:
A. 1 ✓ Correct Answer
B. Less than 1
C. Greater than 1
D. Zero
Explanation: By definition, the total probability represented by the
area under a density curve must equal 1.
Q23. A probability density curve can never go:
A. Below the horizontal
scale ✓ Correct Answer
B. Above the horizontal scale
C. Above the X-axis
D. Zero
Explanation: Since probability density is never negative, the curve
can never dip below the horizontal (X) axis.
Q24. Probability is obtained by calculating the area
under the curve in the case of a:
A. Discrete variable
B. Continuous variable ✓ Correct Answer
C. Discontinuous variable
D. None
Explanation: For continuous variables, probabilities correspond to the
area under the density curve over an interval.
Q25. In the case of a continuous variable, the
probability at a single particular point is:
A. Zero ✓ Correct Answer
B. Negative
C. Greater than 1
D. Less than 1
Explanation: Since a single point has zero width, the area under the
density curve at exactly that point is zero.
Q26. If Y is a continuous random variable, P(Y = a) =
A. Zero ✓ Correct Answer
B. Negative
C. Greater than 1
D. Less than 1
Explanation: For any continuous random variable, the probability of it
taking one exact value is always zero.
Q27. For a continuous random variable, P(-∞ < X
< ∞) =
A. 0
B. 1 ✓ Correct Answer
C. Undefined
D. None
Explanation: The probability that X falls somewhere across its entire
possible range must always equal 1.
Q28. A random variable assuming all possible values
in a range (interval) is called a:
A. Continuous variable ✓ Correct Answer
B. Discrete variable
C. Discontinuous variable
D. None
Explanation: Taking on any value within an interval is exactly what
defines a continuous random variable.
Q29. Probability density functions are typically
presented using a:
A. Smooth curve ✓ Correct Answer
B. Bar chart
C. Histogram
D. None
Explanation: Since continuous variables have infinitely many possible
values, their density is shown as a smooth curve rather than discrete bars.
Q30. The total area under the probability (density)
graph is:
A. 1 ✓ Correct Answer
B. 0
C. Negative
D. None
Explanation: Just like with any valid probability distribution, the
total area under the curve must sum to 1.
Q31. A discrete random variable almost always arises
in connection with:
A. Continuous processes
B. Counting ✓ Correct Answer
C. Both (a) and (b)
D. None
Explanation: Discrete variables typically arise from counting
processes (e.g., number of heads, number of defective items).
Q32. The probability distribution of a continuous
random variable is called its:
A. Probability density
function ✓ Correct Answer
B. Probability mass function
C. Both (a) and (b)
D. None
Explanation: For continuous variables, the distribution is
specifically called the probability density function.
Q33. In any density function, the integral from -∞ to
a specific value is called the:
A. Cumulative distribution
function ✓ Correct Answer
B. Density function
C. Mass function
D. None
Explanation: Integrating the density function up to a point gives the
cumulative distribution function (CDF) at that point.
Q34. F(-∞) =
A. 0 ✓ Correct Answer
B. 1
C. Undefined
D. None
Explanation: The cumulative distribution function starts at 0, since
there's no probability accumulated before negative infinity.
Q35. F(∞) =
A. 0
B. 1 ✓ Correct Answer
C. Undefined
D. None
Explanation: The cumulative distribution function reaches 1 at
positive infinity, since all probability has accumulated by then.
Q36. A distribution (cumulative) function is always:
A. Decreasing
B. Increasing ✓ Correct Answer
C. Negative
D. None
Explanation: A cumulative distribution function is non-decreasing — it
always increases or stays flat as x increases, never decreases.
Q37. F(Y) = ∫f(y)dy from -∞ to y is the:
A. Probability density function
B. Distribution
function ✓ Correct Answer
C. Probability mass function
D. None
Explanation: Integrating the density function from -∞ up to y gives
the cumulative distribution function, F(Y).
Q38. The expectation of the deviation from the mean,
i.e. E[X - E(X)], is:
A. Always zero ✓ Correct Answer
B. Always 1
C. Greater than 1
D. Less than 1
Explanation: Since deviations from the mean average out, the expected
deviation from the mean is always zero.
Q39. The expectation of a random variable is the:
A. Mean of the
distribution ✓ Correct Answer
B. Variance of the distribution
C. Standard deviation of the
distribution
D. None
Explanation: By definition, E(X) represents the mean (average) of the
random variable's distribution.
Q40. If 'c' is a constant, then E(c) =
A. c ✓ Correct Answer
B. Zero
C. Undefined
D. None
Explanation: The expectation of a constant is simply that constant
itself, since it never varies.
Q41. If a and b are constants, then E(ax + b) =
A. aE(x) + b ✓ Correct Answer
B. aE(x)
C. E(x)
D. None
Explanation: By the linearity property of expectation, E(ax+b) =
a·E(x) + b.
Q42. E(5X + 10) = 5E(X) +
A. a
B. 10 ✓ Correct Answer
C. Negative
D. None
Explanation: Applying E(ax+b) = aE(x)+b with a=5, b=10 gives E(5X+10)
= 5E(X) + 10.
Q43. If E(X) = 3, then E[-2X + 5] =
A. 1
B. -1 ✓ Correct Answer
C. 5
D. -6
Explanation: E(-2X+5) = -2×E(X) + 5 = -2(3) + 5 = -6 + 5 = -1.
Q44. If S.D(X) = 2, then S.D[-2X + 5] =
A. -4
B. 4 ✓ Correct Answer
C. 5
D. 9
Explanation: Standard deviation scales with the absolute value of the
multiplier: S.D(-2X+5) = |-2| × S.D(X) = 2 × 2 = 4.
Q45. E(X) = Σx·f(x), provided the sum:
A. Absolutely converges ✓ Correct Answer
B. Absolutely diverges
C. Is undefined
D. None
Explanation: For the expectation formula to be valid, the sum Σx·f(x)
must absolutely converge.
Q46. Var(X) =
A. E{X - E(X)}²
B. E(X²) - [E(X)]²
C. E{X - μ}²
D. All of these ✓ Correct Answer
Explanation: All three expressions are equivalent, standard formulas
for computing the variance of X.
Q47. If X and Y are independent, then Var(X - Y) =
A. Var(X) + Var(Y) ✓ Correct Answer
B. Var(X) - Var(Y)
C. Var(X)·Var(Y)
D. None
Explanation: For independent variables, variances always add
regardless of whether you're looking at the sum or the difference: Var(X-Y) =
Var(X) + Var(Y).
Q48. If 'a' is a constant, then Var(aX) =
A. a² · V(x) ✓ Correct Answer
B. a · V(x)
C. V(x)
D. None
Explanation: Variance scales with the square of the multiplying
constant: Var(aX) = a²·Var(X).
Q49. If V(X) = 4, V(Y) = 3, and X, Y are independent,
then V[X ± Y] =
A. 1
B. 7 ✓ Correct Answer
C. 12
D. Doesn't exist
Explanation: For independent variables, Var(X±Y) = Var(X) + Var(Y) = 4
+ 3 = 7 (variances add regardless of + or − sign).
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