Mock Cup Quiz 1

First Question: Joint Probability Mass Function (PMF)

• Asked: Are X and Y independent?
• Condition: f(X,Y) = f(X) * f(Y)
• Process:
  • Find all marginals of X and Y.
  • If for all elements, f(X,Y) = f(X) * f(Y), then X and Y are independent.
• Conclusion: By checking a single element, it was found that they are not independent.

Second Question: Find the value of P(X+Y = 3)

• Table: Values of X and Y, range 1 and 2
• Possible cases:
  1. X = 1, Y = 2
  2. X = 2, Y = 1
• Process and Result:
  • Added P(1,2) and P(2,1)
  • Answer: 0.625

Third Question: Probability (P) of Max(X, Y) < 2

• Condition: Max(X,Y) < 2
• Possible events:
  1. X < 2 and Y < 2
  2. X = 2 and Y < 2
  3. X < 2 and Y = 2
• Answer: Added all probabilities, answer = 1 (since all possibilities were added).

Fourth Question: Normal Distribution

• Given: μ = 0, σ² = 1 (Standard Normal)
• Asked: P(X < -2)
• Conclusion: Since normal distribution is symmetric, P(X > 2) = 0.1
• Answer: 0.8 (area between -2 and 2)

Fifth Question: Expected Trials to Get Same Value on Both Dice

• Success probability: 1/6
• Failure probability: 5/6
• Geometric distribution: E(X) = 1/p
• Answer: 6

Sixth Question: Discrete to Continuous Variable

• Given: F(x)
• Asked: Find values for continuous variables
• Solving and Final result:
  • Increment the range, probability values
  • Result: Correct answer based on interpretations and range

Seventh Question: Exponential Distribution

• Given: E(X) = 20
• Lambda value: 1/20
• Asked: P(X < 15)
• Process: 1 - e^(-λx) which is 1 - e^(-15/20)
• Answer: 0.53

Eighth Question: Player Rolls Two Dice

• Given: Rules of points
• Asked: Expected earnings
• Answer: 15

Ninth Question: Geometric Distribution

• Given: Binomial distribution
• Asked: P(Y > 5 | X = 2)
• Process: Used conditional probability
• Answer: 0.031

Tenth Question: Height of Students

• Given: μ and σ
• Chebyshev’s inequality: k1/sqrt(k²-1)
• Answer: 1/9

Eleventh Question: PDF of Discrete Variable

• Given: Support and values
• Found: Range (Support)
• Correct answer: 0 < x < 1

Twelfth Question: Bernoulli Trials

• Given: X1 to X10, p = 1/4
• Asked: P(X1=0 | Y=8)
• Sum (y = 8): Binomial
• Answer: 0.2