Distribution

• What is a random variable?

  A variable whose outcome is unknown until measured
• How are random variables denoted?

  By upper case letters like X or Y
• What does P(X = x) represent?

  The probability that random variable X equals x
• What does a probability distribution describe?

  The probability of any outcome in the sample space
• What is a discrete uniform distribution?

  When all probabilities are the same
• What is the random variable defined in Example 1?

  X is the number of heads from three coins

  
• What are the possible outcomes when tossing three fair coins?

  • HHH
  • HHT
  • HTH
  • THH
  • HTT
  • THT
  • TTH
  • TTT
• What is the probability of getting 0 heads when tossing three fair coins?

  1/8
• What is the probability of getting 1 head when tossing three fair coins?

  3/8
• What is the probability of getting 2 heads when tossing three fair coins?

  3/8
• What is the probability of getting 3 heads when tossing three fair coins?

  1/8
• What are the cumulative probabilities defined in the study material?

  • P(X ≤ x): Sum of probabilities up to x
  • P(X < x): Sum of probabilities not greater than x
  • P(X ≥ x): Sum of probabilities for x and greater
  • P(X > x): Sum of probabilities greater than x
• How can you find cumulative probabilities for X~B(n, p)?

  Using tables or a calculator
• What is the probability of landing on red with the spinner?

  0.3
• What is the random variable defined for Jane's spins?

  X is the number of reds in 12 spins
• What is the probability that Jane obtains no more than 2 reds?

  P(X ≤ 2) = 0.2528
• How is the probability of at least 5 reds calculated?

  P(X ≥ 5) = 1 − P(X ≤ 4)
• What is the first value of x that gives a probability greater than 0.95?

  x = 6
• What is the condition for winning a prize in Jane's competition?

  P(X ≥ r) < 0.05
• What is the binomial distribution notation for a random variable?

  X ~ B(n, p)
• What are the conditions for a binomial distribution?

  • Fixed number of trials, n
  • Two possible outcomes (success/failure)
  • Fixed probability of success, p
  • Trials are independent
• What is the probability mass function for a binomial distribution?

  P(X = r) = $\binom{n}{r} p^r (1 - p)^{n - r}$
• What is the probability that exactly 7 members are left-handed?

  P(X = 7) = 0.0160
• How do you calculate the probability of fewer than two left-handed members?

  P(X < 2) = P(X = 0) + P(X = 1)
• What is the total probability of fewer than two left-handed members?

  0.176
• What is the formula for combinations in statistics?

  $nCr =$$\frac{n!}{r!(n - r)!}$