Random Variable
Week 6 Summary (DSME5110F)
• A variable whose value is random (the outcome of a random experiment) • Two types:
– Discrete (binomial, . . . ) – Continuous (normal, . . . )
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Discrete Distribution
• probability mass function
• Expectation, variance and standard deviation • Rules:
– E(aX+b)=aE(X)+b;
– Var(aX + b) = a2Var(X);
– E(X+Y)=E(X)+E(Y);
– If X and Y are independent, then Var(X + Y ) = Var(X) + Var(Y ).
• Two conditions:
– A sequence of independent and identical trials;
– Only two possible outcomes (with probabilities p and 1 − p).
• R functions: dbinom(r, n, p), pbinom(r, n, p), qbinom(y, n, p), rbinom(k, n, p)
• Expectation: np; Variance: np(1 − p); (Recall that we can represent X as X = U1 + U2 + · · · + Un:
Ui = 1 with probability p and Ui = 0 with probability 1 − p)
• Proportion of successes: X = U1+U2+···+Un
– Expectation: p – Variance: p(1−p)
Continuous Distribution
• Probability density function
• probability: area below the density function (for reference only)
• Two parameters: μ: location; σ: shape (dispersion)
• R functions: dnorm(x, μ, σ), pnorm(x, μ, σ), qnorm(p, μ, σ), rnorm(n, μ, σ)
– Default values: μ = 0 and σ = 1 (standard normal)
• X follows normal(μ, σ) if and only if Z := X−μ follows normal(0, 1) (X = μ + σZ)
Point Estimator and Central Limit Theorem
• Sample mean: if n is large, then X ̄ = X1+X2+···+Xn is approximately normally distributed with mean
μ and standard deviation σ/ n.
• Sample proportion (Proportion of successes, use normal to approximate binomial): approximately
normally distributed with mean p and standard deviation p(1−p) n
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