Flashcards · 4040 · Topic 12

Probability distributions flashcards

11 cards on Topic 12, subtopics 12.1 to 12.2. Pick a subtopic or study them all.

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What is a probability distribution of a discrete random variable?: A table of every value x that X can take, with its probability p(x). Capital X is the variable; small x is a value it takes.
What must be true of the probabilities in a distribution?: Each satisfies 0 ≤ p(x) ≤ 1, and together they sum to exactly 1: Σp(x) = 1.
p(0) = 0.1, p(1) = 0.3, p(2) = k and p(3) = 0.35. Find k: k = 1 − (0.1 + 0.3 + 0.35) = 0.25.
A box has 4 red and 6 blue pens. Two are taken without replacement. Find the distribution of X, the number of red pens: p(0) = 6/10 × 5/9 = 1/3. p(1) = 2 × (4/10 × 6/9) = 8/15. p(2) = 4/10 × 3/9 = 2/15. Check: 5/15 + 8/15 + 2/15 = 1.
How do you find two unknown probabilities when E(X) is given?: Form one equation from Σp(x) = 1 and another from Σxp(x) = E(X), then solve them together. Example: p(1) = 0.15, p(2) = a, p(3) = b, p(4) = 0.25, E(X) = 2.7 → a = 0.25, b = 0.35.

How do you calculate E(X), and what is the common mistake?: E(X) = Σ x p(x): multiply each value by its probability and add. Do not divide by the number of values; E(X) is weighted by the probabilities.
What is E(aX + b), and does the rule work for E(X²)?: E(aX + b) = aE(X) + b. It holds only for straight-line functions: E(X²) = Σx²p(x), which is not the same as [E(X)]².
p(1) = 0.4, p(2) = 0.35, p(3) = 0.25. Find E(X), E(X²) and E(3X − 1): E(X) = 0.4 + 0.70 + 0.75 = 1.85. E(X²) = 0.4 + 1.40 + 2.25 = 4.05. E(3X − 1) = 3 × 1.85 − 1 = 4.55.
Stake Rs 3. A fair 1–5 spinner pays Rs 10 for a 5 and Rs 4 for a 4. Find the expected net gain: E(received) = 10 × 1/5 + 4 × 1/5 = 2.8. Expected net gain = 2.8 − 3 = −Rs 0.20 per play, an expected loss of Rs 30 over 150 plays.
What is a fair game?: One where the expected net gain is zero, so the stake equals the expected amount received. Example: Rs 1 to roll a die, prize for a 6 → the game is fair if the prize is Rs 6.
A raffle of 500 tickets has prizes totalling Rs 500. What ticket price makes it fair?: Expected prize per ticket = 500 ÷ 500 = Rs 1, so a Rs 1 ticket is fair. At Rs 1.50 the organiser expects Rs 0.50 per ticket.