Binomial Distribution Quiz

Binomial Distribution and Parameter Determination

• For a Binomial distribution with parameters ( n=2 ) and ( P(0)=0.7056 ), the probability of getting 0 successes can be expressed as ( P(0) = (1-p)^n ).
• Plugging in the values gives ( P(0) = (1-p)^2 = 0.7056 ).
• Solving for ( p ), we find ( 1-p = \sqrt{0.7056} ) resulting in ( 1-p \approx 0.84 ), so ( p \approx 0.16 ).

Binomial Distribution: Usual Values for Random Variable

• The mean ( \mu ) of a Binomial distribution is calculated as ( \mu = n \cdot p ).
• Given ( \mu = 250 ) and ( \sigma = 11.18 ), we can derive ( p ) and ( n ) if needed.
• The usual value, typically denoted as 2 standard deviations from the mean, can be calculated as:
  • Minimum usual value = ( \mu - 2\sigma = 250 - 2(11.18) ).
  • This results in a minimum usual value of approximately ( 250 - 22.36 = 227.64 ).

Evaluating Probability Distribution Table

• To determine if a table represents a valid probability distribution:
  • All probabilities must be between 0 and 1 inclusive.
  • The sum of all probabilities must equal 1.
• Assess the table by checking these criteria.

Probability Distribution Table Variable ( k )

• When calculating ( k ) in a probability distribution table, it is required that the total probability sums to 1.
• Set up an equation based on the values in the table, ( k + P_1 + P_2 + \ldots = 1 ).
• Solve for ( k ) to determine its value.

Expected Value Calculation

• The expected value ( E(X) ) for discrete variables is computed using the formula ( E(X) = \sum (x_i \cdot P(x_i)) ).
• For the random variable ( X ) with values 2, 3, 4 and probabilities 0.3, 0.3, 0.4:
  • ( E(X) = (2 \cdot 0.3) + (3 \cdot 0.3) + (4 \cdot 0.4) ).
  • Calculation yields ( E(X) = 0.6 + 0.9 + 1.6 = 3.1 ), confirming that the statement is true.