Probability and Random Variables Quiz
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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?

  • 1/6
  • 1/12 (correct)
  • 1/3
  • 1/4

What is the probability of an impossible event?

  • 1/4
  • 1/2
  • 1
  • 0 (correct)

For the probability density function 𝑓(𝑥) = 𝑘𝑥, where 0 < 𝑥 < 1, what is the value of k?

  • 1 (correct)
  • 2
  • 0
  • 3

If the discrete random variable X has values (0, 𝑘), (1, 2𝑘), (2, 3𝑘), (3, 4𝑘), what is the value of k?

<p>0.1 (A)</p> Signup and view all the answers

If events A and B are mutually exclusive, what is the probability P(A|B)?

<p>0 (C)</p> Signup and view all the answers

What is the formula for Cov(aX, bY) when X and Y are random variables?

<p>abCov(X,Y) (D)</p> Signup and view all the answers

If X and Y are independent random variables, what is the value of Cov(X,Y)?

<p>0 (D)</p> Signup and view all the answers

What is the mean of a binomial distribution with parameters n and p?

<p>np (B)</p> Signup and view all the answers

The variance of a binomial distribution is given by which of the following formulas?

<p>npq (C)</p> Signup and view all the answers

What is the standard deviation of a binomial distribution with parameters n and p?

<p>√npq (C)</p> Signup and view all the answers

What is the probability of getting exactly 3 heads when tossing 6 coins?

<p>0.2325 (A)</p> Signup and view all the answers

Which distribution is considered the limiting case of the binomial distribution?

<p>Poisson distribution (A)</p> Signup and view all the answers

For a standard normal distribution, what is the mean?

<p>0 (D)</p> Signup and view all the answers

What is the standard deviation of a Poisson distribution with parameters n=10000 and p=0.001?

<p>3.1622 (A)</p> Signup and view all the answers

If α = 1/3 in an exponential distribution, what is the mean?

<p>1/√3 (C)</p> Signup and view all the answers

What is the variance of an exponential distribution when α = 1/2?

<p>1/4 (A)</p> Signup and view all the answers

What is the variance of a Poisson distribution with parameter λ = 2?

<p>2 (A)</p> Signup and view all the answers

In a Poisson distribution, if 2P(X = 1) = P(X = 2), what is the variance?

<p>2 (C)</p> Signup and view all the answers

If P(X = 2) = P(X = 3) in a Poisson distribution, what is P(X = 0)?

<p>e^-3 (D)</p> Signup and view all the answers

In a binomial distribution, which characteristic does NOT apply?

<p>Each trial results in two or more outcomes. (B)</p> Signup and view all the answers

What is the relationship between the mean and variance in a binomial distribution when p = 0.5?

<p>They are only equal when p = 0.5. (D)</p> Signup and view all the answers

What is the variance of the expression $V(19-4X)$ if $V(X) = 7.15$?

<p>114.4 (A)</p> Signup and view all the answers

Given $E[X] = 3$ and $E[X^2] = 10$, what is the variance $V(X)$?

<p>7 (A)</p> Signup and view all the answers

If $E[X] = 2$ and $E[X^2] = 12$, what is the standard deviation of $X$?

<p>3.1622 (C)</p> Signup and view all the answers

If $V(X) = 15$, what is the variance of the expression $V(8X)$?

<p>120 (B)</p> Signup and view all the answers

If $A$ and $B$ are mutually exclusive events, what is $P(A igcap B)$?

<p>0 (A)</p> Signup and view all the answers

Given that $E[X] = 3$ and $V[X] = 18$, what is $E[X^2]$?

<p>27 (C)</p> Signup and view all the answers

If $X$ and $Y$ are independent random variables, what is $E(XY)$?

<p>E(X)E(Y) (D)</p> Signup and view all the answers

What is the marginal distribution of $X$ given the joint pdf $f(x, y) = e^{-(x + y)}$ for $0 < x, y < ∞$?

<p>e^{-x} (B)</p> Signup and view all the answers

Flashcards

Conditional Probability

The probability of an event A occurring given that another event B has already occurred. It is calculated by dividing the probability of both events happening by the probability of the event that has already occurred.

Mutually Exclusive Events

Events that cannot occur at the same time. If one event occurs, the other cannot.

Independent Events

Events where the occurrence of one event does not affect the probability of the other event occurring.

Probability of an Impossible Event

The probability of an event that cannot occur is always zero.

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Probability Density Function (PDF)

A function that describes the probability distribution of a continuous random variable. It represents the relative likelihood of a random variable taking on a specific value.

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V(19-4X)

The variance of a linear transformation of a discrete random variable X. The formula is: V(aX + b) = a²V(X), where 'a' and 'b' are constants.

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Variance of a Variable

The variance (V(X)) of a random variable X measures the spread or dispersion of its values around the mean. It is calculated as the expected value of the squared deviations from the mean: V(X) = E[(X - E(X))²].

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Standard Deviation

The standard deviation (σ) of a random variable X is the square root of its variance (V(X)). It provides a measure of the average deviation of the variable's values from the mean.

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E(XY)

The expected value of the product of two independent random variables X and Y is equal to the product of their individual expected values: E(XY) = E(X)E(Y).

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V(X-Y)

For independent random variables X and Y, the variance of their difference is the sum of their individual variances: V(X - Y) = V(X) + V(Y).

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Marginal Distribution

The marginal distribution of a variable in a joint probability distribution represents the probabilities of that variable regardless of the value of the other variable(s).

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Covariance of Transformed Variables

The covariance of two random variables multiplied by constants is equal to the product of the constants and the original covariance.

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Covariance of Independent Variables

The covariance of two independent random variables is always zero.

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Binomial Mean

The expected value of a binomial distribution is the product of the number of trials (n) and the probability of success (p).

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Standard Normal Distribution Symmetry

The probability density function (PDF) of a standard normal distribution is symmetric about the y-axis and the x-axis.

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Binomial Standard Deviation

The standard deviation of a binomial distribution is the square root of the product of the number of trials (n), the probability of success (p), and the probability of failure (q).

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Binomial Variance

The variance of a binomial distribution is the product of the number of trials (n), the probability of success (p), and the probability of failure (q).

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Probability of 'k' Successes

The probability of 'k' successes in a binomial distribution with 'n' trials and probability of success 'p' is given by the formula: nCk * p^k * q^(n-k).

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Poisson Distribution

The probability of a certain number of events occurring in a given time or space when events occur independently and at a constant rate.

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Poisson Standard Deviation

For a Poisson distribution with mean (λ) equal to np, where n is the number of trials and p is the probability of success, the standard deviation is the square root of the mean.

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Exponential Distribution Mean

In an exponential distribution with parameter α, the mean is represented by 1/α. This represents the average time or duration before an event occurs.

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Exponential Distribution Variance

In an exponential distribution with parameter α, the variance is represented by 1/α². It measures how spread out the data is around the mean.

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Poisson Variance

In a Poisson distribution, the variance is equal to the mean (λ). This means that the spread of the distribution is directly proportional to the average number of events.

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Poisson Distribution Relationship

In a Poisson distribution, the relationship between P(X=1) and P(X=2) can be used to find the variance. If 2*P(X=1) = P(X=2), the variance equals 2.

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Poisson Probability

In a Poisson distribution with a mean of λ, the probability of having 0 events is given by e^(-λ).

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Binomial Distribution Characteristic

The binomial distribution is characterized by independent trials where each trial results in only two possible outcomes: success or failure. The probability of success remains constant across all trials.

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Poisson Probability with Mean

For a Poisson distribution with mean λ, the probability of observing X or more events can be calculated using the formula 1 - ΣP(X = i) for i = 0 to X-1. This represents the probability of having at least X events.

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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

  • If E[X] = 2, E[X²] = 12, then the standard deviation is 2.8284.

  • If V(X)= 15, then V(8X) is 960

  • If E[X]= 3 , V(X) is 18 then E(X^2) is 27

  • If P(0 < X < 0.5) and pdf of random variable is 2x, 0<x<1, is 0.25

  • The mean of a binomial distribution is np; If n = 10000, p = 0.001, then standard deviation is 3.1622.

  • Mean of exponential distribution with a = 1/3 is 3

  • Variance of exponential distribution with a = 1/2 is 4.

  • Variance of a Poisson distribution with parameter λ = 2 is 2

  • In Poisson distribution, if 2P(X = 1) = P(X = 2), then variance is 2

  • Standard deviation of binomial distribution is √npq

  • Probability of getting exactly 3 heads with 6 coins is 0.3125

  • 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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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.

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