Probability distributions flashcards

Topic 8: Probability distributions

What is a probability distribution?

A list, usually a table, of every value x that the random variable X can take, with P(X = x) for each value.

What two conditions must a probability distribution meet?

Every probability satisfies 0 ≤ P(x) ≤ 1, and the probabilities add to 1: ΣP(x) = 1.

P(X = x) = kx for x = 1, 2, 3, 4, 5. Find k

k + 2k + 3k + 4k + 5k = 15k = 1, so k = 1/15.

How do you calculate the expectation E(X)?

E(X) = Σ x P(x): multiply each value by its probability and add. Example: values 1, 2, 3 with probabilities 0.2, 0.5, 0.3 → 0.2 + 1 + 0.9 = 2.1.

How many successes do you expect in n independent trials with probability p?

Expected number = n × p. Example: 80 trials with p = 0.35 → 28.

What is the expected gain in a game with a stake?

Expected gain = expected payout − stake. Example: pay £2 to win £5 with probability 0.3 → 5 × 0.3 − 2 = −£0.50, an expected loss of 50p per game.

What makes a game fair?

The expected gain is zero, so the stake equals the expected payout. Example: £8 paid with probability 1/4 → fair stake = 8 × 1/4 = £2.

X is the number of heads in three tosses of a fair coin. Give its distribution and E(X)

P(0) = 1/8, P(1) = 3/8, P(2) = 3/8, P(3) = 1/8. E(X) = (0 + 3 + 6 + 3) ÷ 8 = 1.5.

Must E(X) be a value that X can take?

No. It is the long-run average over many trials, such as 1.5 heads in three tosses. It is also not the most likely value (the mode).