The difference between an estimate and the parameter is called : | Agrigyan

Analysis of degrees of freedom in RBD:
Total number of experimental units is $N = n \times r$.

• Incorrect: $r - 1$ is the degrees of freedom for Replications (Blocks).
• Incorrect: $n - 1$ is the degrees of freedom for Treatments.
• Incorrect: $n \times r - 1$ is the Total degrees of freedom.
• Correct: In a Randomized Block Design (RBD) with $n$ treatments and $r$ replications:
  $$\text{Total df} = nr - 1$$
  $$\text{Block df} = r - 1$$
  $$\text{Treatment df} = n - 1$$
  $$\text{Error df} = (nr - 1) - (r - 1) - (n - 1) = (n - 1)(r - 1)$$

Calculation of probability:

• Incorrect: The event has positive probability.
• Incorrect: It is not a certain event.
• Incorrect: $0.25$ is the probability of getting two heads (HH) or two tails (TT).
• Correct: When two coins are tossed, the sample space is $S = \{HH, HT, TH, TT\}$ with total outcomes $n(S) = 4$. The event of getting one head and one tail corresponds to outcomes $\{HT, TH\}$, so $n(E) = 2$. Therefore, $P(E) = \frac{2}{4} = 0.5$.

Calculation of variance of binomial distribution:
Given $n = 10$, probability of failure $q = 0.3$.
Probability of success is $p = 1 - q = 1 - 0.3 = 0.7$.

• Incorrect: 21 misses the decimal point.
• Correct: The variance of a binomial distribution is given by:
  $$\text{Var}(X) = n \cdot p \cdot q = 10 \times 0.7 \times 0.3 = 2.1$$
• Incorrect: 7 is the mean of the distribution ($np = 10 \times 0.7 = 7$).
• Incorrect: 3 corresponds to $nq = 10 \times 0.3 = 3$.

Analysis of mode in Poisson distribution:
In a Poisson distribution with parameter $\lambda$:

• Correct: When $\lambda$ is an integer (say $k$), the distribution is bimodal, and the two modes occur at values $x = k - 1$ and $x = k$, where $P(X = k - 1) = P(X = k)$.
• Incorrect: The distribution is unimodal only when $\lambda$ is not an integer, with the unique mode at $[\lambda]$ (integral part of $\lambda$).
• Incorrect: 'Positive modal' is not standard statistical terminology.
• Incorrect: 'Negative modal' is not standard statistical terminology.