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Questions and Answers
The expected value of a random variable is analogous to which statistical measure?
The expected value of a random variable is analogous to which statistical measure?
- Mode
- Median
- Range
- Weighted average (correct)
If the sum of $|x_i|p(x_i)$ diverges, then the expected value is equal to infinity.
If the sum of $|x_i|p(x_i)$ diverges, then the expected value is equal to infinity.
False (B)
Expected value is also referred to as the ______ of X.
Expected value is also referred to as the ______ of X.
mean
Which of the following is NOT an equivalent term for 'expected value of X'?
Which of the following is NOT an equivalent term for 'expected value of X'?
The variance of X is equal to E[(X – μ)²], where μ represents the ______ of X.
The variance of X is equal to E[(X – μ)²], where μ represents the ______ of X.
The standard deviation of X is the square of the variance.
The standard deviation of X is the square of the variance.
What does μ'₁ represent?
What does μ'₁ represent?
What must be true about a sequence of real numbers $a_0, a_1,... $ for it to have a generating function G(s)?
What must be true about a sequence of real numbers $a_0, a_1,... $ for it to have a generating function G(s)?
For what series is the following equation true? $1 + s + s^2 + ... = \frac{1}{1-s}$
For what series is the following equation true? $1 + s + s^2 + ... = \frac{1}{1-s}$
Match the generating functions with the sequences they generate:
Match the generating functions with the sequences they generate:
In probability theory, for what case is the generating function G(s) particularly useful?
In probability theory, for what case is the generating function G(s) particularly useful?
If the distribution of sums, $\sum_{i=1}^{n} X_i$, is independent then the generating function is easy to handle.
If the distribution of sums, $\sum_{i=1}^{n} X_i$, is independent then the generating function is easy to handle.
What is the abbreviation of 'Probability Generating Functions'?
What is the abbreviation of 'Probability Generating Functions'?
For what type of random variables are Probability Generating Functions useful?
For what type of random variables are Probability Generating Functions useful?
Suppose X is a random variable with pmf $p_X(x)$, that is P(X = x) = $p_X(x)$; X=0,1,2,3,... Then $P_X(s) = \sum_{x=0}^{\infty} p_X(x)s^x$ is the _______.
Suppose X is a random variable with pmf $p_X(x)$, that is P(X = x) = $p_X(x)$; X=0,1,2,3,... Then $P_X(s) = \sum_{x=0}^{\infty} p_X(x)s^x$ is the _______.
$\sum_{x=0}^{\infty} p_X(x) = 1$ given that $P_X(1)$ exists for |S| ≤ 1.
$\sum_{x=0}^{\infty} p_X(x) = 1$ given that $P_X(1)$ exists for |S| ≤ 1.
How are moments determined by a probability generating function?
How are moments determined by a probability generating function?
V(X) = $E(X^2) – [E(X)]^2$ = E[X(X – 1)] + E(X) – [E(X)]2. represents the what adjustment?
V(X) = $E(X^2) – [E(X)]^2$ = E[X(X – 1)] + E(X) – [E(X)]2. represents the what adjustment?
X follows various distributions. Match the distribution with its parameters:
X follows various distributions. Match the distribution with its parameters:
Assume $X \sim Poisson(\mu)$. What is the probability generating function, $P_X(s)$?
Assume $X \sim Poisson(\mu)$. What is the probability generating function, $P_X(s)$?
Flashcards
Expected Value
Expected Value
The expected value of a random variable, much like a weighted average.
Frequency Function
Frequency Function
A function, denoted p(x), that gives the probability of a discrete random variable taking on a specific value.
Mean (μ)
Mean (μ)
The mean of X, often represented by μ. It's the long-run average value of X.
Variance
Variance
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Standard Deviation
Standard Deviation
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Generating Function
Generating Function
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Probability Generating Function (PGF)
Probability Generating Function (PGF)
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Binomial Distribution
Binomial Distribution
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Poisson Distribution
Poisson Distribution
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Study Notes
- The concept of the expected value of a random variable mirrors the weighted average.
- Values of the random variable are weighted by probabilities in the definition.
- The expected value of X denoted E(X) is E(X) = ∑x_i p(x_i) for a discrete random variable with frequency function p(x).
- The sum ∑|x_i|p(x_i) must be less than ∞, otherwise the expectation is undefined.
Mean and Expected Value
- E(X) is referred to as the mean of X, denoted by μ or μ_X.
- The population mean, μ, or expected value of X, signifies the long-run average value of X.
- E(X) = ∑x_i p(x_i), provided ∑ is absolutely convergent for discrete X.
- E(X) = ∫x f(x) dx, provided ∫ is absolutely convergent for continuous X.
- E[g(X)] = ∑ g(x)p(x) for discrete X
- E[g(X)] = ∫ g(x)f(x) dx for continuous X
Variance and Moments
- A key measure is the variance X, where E[(X − μ)²] = E(X²) − μ² = Var(X) = V(X) = σ².
- The standard deviation of X is the square root of the variance, σ = √σ².
- Moments of X are of interest.
- The rth moment of X about the origin (0) is μr = E(X^r).
- The rth moment of X about the mean (μ), also the rth central moment is μr = E[(X − μ)^r].
- μ'₁ = E(X) = μ, μ₁ = E[(X − μ)] = 0, μ₂ = E[(X − μ)²] = σ²
Generating Functions
- Generating functions offer a way to express distributions.
- Given a sequence of real numbers a₀, a₁, ..., {ar}, the generating function is G(s) = a₀ + a₁s + a₂s² + ... = ∑ ar s^r.
- G(s) = 1/(1-s) generates 1, 1, 1, 1,... because 1 + s + s² + ... = 1/(1-s) (geometric series for |s| < 0).
- G(s) = 1/(1+s) generates 1, -1, 1, -1,... because 1 – s + s² – s³ + ... = ∑(-1)^r s^r = 1/(1+s).
- G(s) = 1/(1-s)² generates 1, 2, 3,...
Generating Functions Use Cases
- G(s) compactly represents a sequence.
- In probability theory, a_r = P(X = r) in the discrete case or a_r is the rth moment of X.
- G(s) can easily facilitate handling to obtain:
- A complete sequence of probabilities
- Distributions of linear transformations aX + b.
- Distributions of sums, ∑ Xᵢ, where the Xᵢ are independent.
- Distributions of random sums ∑Xᵢ.
Probability Generating Functions (PGFs)
- Probability Generating Functions (PGFs) apply to discrete random variables.
- Assume X is defined for 0, 1, 2,...
- For a random variable X with probability mass function pX(x), P(X = x) = pX(x) where X = 0,1,2,3,... and the PGF is Px(s) = ∑ pX(x)s^x.
- Px(s) generates {pX(x)}.
- Px(1) = ∑ pX(x) = 1, and Px(1) exists for |S| ≤ 1.
- From the definition, Px(s) = E[S^X].
- Moments can be found by successive differentiation, with adjustments.
Standard Moments
- P'x(1) = dPx(S)/dS |s=1
- P'x(1) = ∑ xpX(x)s^(x-1) |s=1 = ∑ xpX(x) = E(X)
- P"x(1) = ∑ x(x − 1)pX(x)s^(x-2) |s=1 = E[X(X − 1)]
- Variance is adjusted as V(X) = E(X²) – [E(X)]² = E[X(X – 1)] + E(X) – [E(X)]².
- Therefore, V(X) = P"x(1) + P'x(1) – [P'x(1)]².
Theorems and Examples
- If X ~ Bin(n, θ), then
- Px(s) = ∑ (n choose x) θ^x (1 − θ)^(n−x) s^x = (1 − θ)^n ∑ (n choose x) (Sθ/(1-θ))^x = (1 − θ)^n [1 + Sθ/(1-θ)]^n.
- Px(s) = ((1 − θ) + Sθ)^n
- If X ~ Poisson(μ), then
- Px(s) = ∑ e^(-μ) μ^x s^x / x! = e^(-μ) ∑ (sμ)^x / x! = e^(-μ) e^(sμ) = e^((s-1)μ).
Exercises
- If X ~ P(λ), derive:
- Px(s)
- E(X)
- Var(X)
- If X ~ Bin(n, θ), derive:
- Px(s)
- E(X)
- Var(X)
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