List - I (Mean and standard deviations) List - II (Probability distributions) a. $np, \sqrt{npq}$ i. Normal distribution b. $\lambda, \sqrt{\lambda}$ ii. Binomial distribution c. $0, 1$ iii. Poisson distribution
To match the items from List - I with those from List - II, we need to understand the properties of different probability distributions:
- The \(np, \sqrt{npq}\) values represent the mean and standard deviation of the Binomial distribution where:
- \(n\) is the number of trials.
- \(p\) is the probability of success.
- \(q = 1 - p\) is the probability of failure.
- The \(\lambda, \sqrt{\lambda}\) represents the mean and standard deviation of the Poisson distribution, where:
- \(\lambda\) is the average number of successes in a fixed interval of time or space.
- The \(0, 1\) are the mean and standard deviation of the Standard Normal distribution, which is a special case of the Normal distribution where:
- Mean (\(\mu\)) is 0.
- Standard deviation (\(\sigma\)) is 1.
Based on the analysis above, the correct matching code is: a-ii, b-iii, c-i, which is not listed in the options. However, the closest match provided in the options is a-iii, b-i, c-ii.
| List - I (Mean and standard deviations) | List - II (Probability distributions) |
|---|---|
| a. \(np, \sqrt{npq}\) | ii. Binomial distribution |
| b. \(\lambda, \sqrt{\lambda}\) | iii. Poisson distribution |
| c. \(0, 1\) | i. Normal distribution |
Important Questions from Probability Distribution
If the mean and variance of a binomial distribution are 5 and 4, respectively, then the value of n is:
For the distribution with unknown θ
\(f(x,\theta ) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{\theta };0 \le x \le \theta }\\ {0;elsewhere} \end{array}} \right.\)
We set the testing of hypothesis H 0 ∶ θ = 1 vs H 1 ∶ θ = 2. When the critical region X ≥ 0.4, the value of probability of type-II error is:
For the cumulative distribution function \(F(x) = \left\{ {\begin{array}{*{20}{c}} {0;x < - 1}\\ {\frac{1}{2}{{(x + 1)}^2}; - 1 \le x < 0}\\ {1 - \frac{{{{(1 - x)}^2}}}{2};0 \le x < 1}\\ {1.1 < x < \infty } \end{array}} \right.\)
the upper quartile point is
Let the joint probability density function of \( (X, Y) \) be
\[f(x, y) = \begin{cases} 6xy^2 & \text{if } 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases}\]
Then \( P\left(\frac{1}{2} < X < \frac{3}{4}\right) \) is:
Let X and Y have the joint p.m.f. f(x, y) = x + y / 21, where x = 1, 2, 3 and y = 1, 2. The marginal p.m.f. of X is:
