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Questions and Answers
What is the value of 𝑃(𝐴|𝐵) if 𝑃(𝐴) = 1/4, 𝑃(𝐵) = 1/3, and 𝑃(𝐴 ∪ 𝐵) = 1/2?
What is the value of 𝑃(𝐴|𝐵) if 𝑃(𝐴) = 1/4, 𝑃(𝐵) = 1/3, and 𝑃(𝐴 ∪ 𝐵) = 1/2?
What is the probability of an impossible event?
What is the probability of an impossible event?
For the probability density function 𝑓(𝑥) = 𝑘𝑥, where 0 < 𝑥 < 1, what is the value of k?
For the probability density function 𝑓(𝑥) = 𝑘𝑥, where 0 < 𝑥 < 1, what is the value of k?
If the discrete random variable X has values (0, 𝑘), (1, 2𝑘), (2, 3𝑘), (3, 4𝑘), what is the value of k?
If the discrete random variable X has values (0, 𝑘), (1, 2𝑘), (2, 3𝑘), (3, 4𝑘), what is the value of k?
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If events A and B are mutually exclusive, what is the probability P(A|B)?
If events A and B are mutually exclusive, what is the probability P(A|B)?
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What is the formula for Cov(aX, bY) when X and Y are random variables?
What is the formula for Cov(aX, bY) when X and Y are random variables?
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If X and Y are independent random variables, what is the value of Cov(X,Y)?
If X and Y are independent random variables, what is the value of Cov(X,Y)?
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What is the mean of a binomial distribution with parameters n and p?
What is the mean of a binomial distribution with parameters n and p?
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The variance of a binomial distribution is given by which of the following formulas?
The variance of a binomial distribution is given by which of the following formulas?
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What is the standard deviation of a binomial distribution with parameters n and p?
What is the standard deviation of a binomial distribution with parameters n and p?
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What is the probability of getting exactly 3 heads when tossing 6 coins?
What is the probability of getting exactly 3 heads when tossing 6 coins?
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Which distribution is considered the limiting case of the binomial distribution?
Which distribution is considered the limiting case of the binomial distribution?
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For a standard normal distribution, what is the mean?
For a standard normal distribution, what is the mean?
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What is the standard deviation of a Poisson distribution with parameters n=10000 and p=0.001?
What is the standard deviation of a Poisson distribution with parameters n=10000 and p=0.001?
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If α = 1/3 in an exponential distribution, what is the mean?
If α = 1/3 in an exponential distribution, what is the mean?
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What is the variance of an exponential distribution when α = 1/2?
What is the variance of an exponential distribution when α = 1/2?
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What is the variance of a Poisson distribution with parameter λ = 2?
What is the variance of a Poisson distribution with parameter λ = 2?
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In a Poisson distribution, if 2P(X = 1) = P(X = 2), what is the variance?
In a Poisson distribution, if 2P(X = 1) = P(X = 2), what is the variance?
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If P(X = 2) = P(X = 3) in a Poisson distribution, what is P(X = 0)?
If P(X = 2) = P(X = 3) in a Poisson distribution, what is P(X = 0)?
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In a binomial distribution, which characteristic does NOT apply?
In a binomial distribution, which characteristic does NOT apply?
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What is the relationship between the mean and variance in a binomial distribution when p = 0.5?
What is the relationship between the mean and variance in a binomial distribution when p = 0.5?
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What is the variance of the expression $V(19-4X)$ if $V(X) = 7.15$?
What is the variance of the expression $V(19-4X)$ if $V(X) = 7.15$?
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Given $E[X] = 3$ and $E[X^2] = 10$, what is the variance $V(X)$?
Given $E[X] = 3$ and $E[X^2] = 10$, what is the variance $V(X)$?
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If $E[X] = 2$ and $E[X^2] = 12$, what is the standard deviation of $X$?
If $E[X] = 2$ and $E[X^2] = 12$, what is the standard deviation of $X$?
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If $V(X) = 15$, what is the variance of the expression $V(8X)$?
If $V(X) = 15$, what is the variance of the expression $V(8X)$?
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If $A$ and $B$ are mutually exclusive events, what is $P(A igcap B)$?
If $A$ and $B$ are mutually exclusive events, what is $P(A igcap B)$?
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Given that $E[X] = 3$ and $V[X] = 18$, what is $E[X^2]$?
Given that $E[X] = 3$ and $V[X] = 18$, what is $E[X^2]$?
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If $X$ and $Y$ are independent random variables, what is $E(XY)$?
If $X$ and $Y$ are independent random variables, what is $E(XY)$?
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What is the marginal distribution of $X$ given the joint pdf $f(x, y) = e^{-(x + y)}$ for $0 < x, y < ∞$?
What is the marginal distribution of $X$ given the joint pdf $f(x, y) = e^{-(x + y)}$ for $0 < x, y < ∞$?
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Study Notes
Probability and Random Variables
- Conditional probability: If P(A) = 1/4, P(B) = 1/3, P(A ∪ B) = 1/2, then P(A|B) = 1/6
- Probability of an impossible event is zero.
- Probability density function (pdf) of a random variable X is f(x) = kx for 0 < x < 1, then k = 2.
- Discrete random variable X with values (0, 1, 2, 3) and probabilities (k, 2k, 3k, 4k) has k = 0.1.
- For events A and B, if P(A) = 0.4, P(B) = 0.04, and P(A ∩ B) = 0.01, then P(A|B) = 0.25
- Mutually exclusive events A and B have P(A ∩ B) = 0.
- Independent events A and B have P(A|B) = P(A).
- If V(X) = 7.15, then V(19 - 4X) = 114.4
- If E[X] = 3 and E[X²] = 10, then V(X) = 7.
Other Statistical Concepts
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If E[X] = 2, E[X²] = 12, then the standard deviation is 2.8284.
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If V(X)= 15, then V(8X) is 960
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If E[X]= 3 , V(X) is 18 then E(X^2) is 27
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If P(0 < X < 0.5) and pdf of random variable is 2x, 0<x<1, is 0.25
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The mean of a binomial distribution is np; If n = 10000, p = 0.001, then standard deviation is 3.1622.
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Mean of exponential distribution with a = 1/3 is 3
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Variance of exponential distribution with a = 1/2 is 4.
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Variance of a Poisson distribution with parameter λ = 2 is 2
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In Poisson distribution, if 2P(X = 1) = P(X = 2), then variance is 2
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Standard deviation of binomial distribution is √npq
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Probability of getting exactly 3 heads with 6 coins is 0.3125
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The moment generating function of uniform distribution is ((e^(at) - e^(-at))/a.
Expected Values
- If X and Y are independent random variables, then E(XY) = E(X)E(Y).
- If the marginal distribution of X is given as (0, 0.4), (1, 0.2), (2, 0.1), (3, 0.3), then E(X) = 1.3 and E[X²]=1.7
- If the joint pdf of X and Y is e^(x+y),0<x,y<∞ ,then marginal distribution of X = ex
Additional Concepts
- The mean of Poisson distribution equals the mean of the exponential distribution when λ = a = 1
- The exponential distribution is defined over the interval [0, ∞].
- Binomial distribution has equal mean and variance only when p = 0.5).
- If X ~ N(5, 32), then P(X < 8) is equals to P(Z < 0.5303)..
- Standard normal distribution has a mean of 0, and a variance of 1, and a standard deviation of 1,
- Variance of a uniform distribution is (b-a)^2/12.
- If X is a sample mean and μ is the population mean, then E(X) = μ.
- Variance of sampling distribution of sample means is σ²/n, where σ is the population standard deviation and n is the sample size.
- If x is a normal variate and X/σ = Y+6 where Y~N(0,1) then mean and variance of x are 6 and 4, respectively.
- Correlation coefficients lies between -1 and 1
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Description
Test your knowledge on key concepts of probability and random variables, including conditional probability, probability density functions, and properties of discrete random variables. This quiz covers essential calculations and theoretical understandings needed for mastering the subject.
