high school Probability Distribution

Probability Distribution

A random variable turns the result of an experiment into a number, like the pips that come up when you roll a die. List the probability of each number in a table and you have a probability distribution, whose center of mass is exactly the expected value. How spread out it is gets measured by variance and standard deviation, and counting successes over many independent trials gives the binomial distribution B(n, p). Move the number of die faces and the expected value (the balance point) sits; change the trial count n and success probability p and the peak of the bar graph shifts.

Random Variables & Distributions

📊 What is a Probability Distribution?

①Random variable: numerical assignment to each outcome

②Distribution: probability of each value, given as a table or function

③Probabilities in a distribution table sum to 1

Expected Value & Variance

Faces on a die6

Expected Value

E(X) = Σ x_i × P(X = x_i)

Mean of a random variable (center of mass)

Variance

V(X) = E(X²) − {E(X)}²

Mean of squared spread

Standard Deviation

σ(X) = √V(X)

Positive square root of variance

Binomial Distribution B(n, p)

Trials n6

Success p × 105

Binomial Probability

P(X = k) = _nC_k p^k (1-p)^n-k

Probability of k successes in n independent trials. Same formula as the previous page. Here k is that page's r

Binomial Mean & Variance

E(X) = np, V(X) = np(1−p)

For B(n, p). When n = 1, V = p(1−p)

Properties of Distributions

Linearity of Expectation

E(aX + b) = aE(X) + b

Constants pass through linearly

Variance Property

V(aX + b) = a²V(X)

Variance measures distance from the mean, so b vanishes and a is squared

Work It Out

Example 1

A random variable X has P(X=0)=1/4, P(X=1)=1/2, P(X=2)=1/4. Find the expected value E(X).

The expected value multiplies each value by its probability and sums.

E(X) = Σ x · P(X = x)

Substitute.

= 0·(1/4) + 1·(1/2) + 2·(1/4) = 1

▸ E(X) = 1

The expected value is a weighted average of the values by their probabilities.

Example 2

For the same X, find the variance V(X). (E(X) = 1)

V(X) = E(X²) − {E(X)}². First find E(X²).

E(X²) = 0²·(1/4) + 1²·(1/2) + 2²·(1/4) = 3/2

Substitute.

V(X) = 3/2 − 1² = 1/2

▸ V(X) = 1/2

The variance subtracts the square of the mean from E(X²).

Wrap-up

Core

P(X=k) = _nC_k p^k(1-p)^n-k, E(X)=np

Binomial probability and expected value

KICE mock-exam Math (Probability & Statistics) type

For a random variable X with E(X) = 3, what is E(2X + 1)?

①5

②6

③7

④8

⑤9

▸ ③ 7

Use the linearity of expectation.

E(aX + b) = aE(X) + b

Substitute.

= 2·3 + 1 = 7

🎯 Exam Points

①Probability table: probabilities sum to 1

②E(X) = Σx·P(x), V(X) = E(X²) − {E(X)}²

③Binomial: B(n,p) → E = np, V = np(1−p)

④E(aX+b) = aE(X)+b

⑤V(aX+b) = a²V(X)

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Independent Trials

Normal Distribution

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