Consider the following with respect to probability distribution:
(A) In a given sample space not more than four random variables can be defined
(B) Mean expectation value is a parameter and not a statistic
(C) In a Gaussian distribution if only mean and mode are same it is said to be normally distributed
(D) Probability of r successes in n Bernoulli trials is \(P(r)={}^{n}C_{r}\,q^{r}p^{n-r}\)
(E) Variance enables us to study the variability of a random variable from the mean expectation
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (E) only
A random variable is any real-valued function on the sample space, so any number of them can be defined. The expected value E(X) describes the whole distribution, so it is a parameter. In a normal distribution mean, median and mode are all equal. With p the probability of success and q = 1 − p,
\(P(r) = {}^{n}C_{r}\,p^{r}q^{n-r}\)
So random variables are unlimited in number, the mean is a parameter, normality needs mean = median = mode, the binomial term is \({}^{n}C_{r}\,p^{r}q^{n-r}\), and variance measures spread about the mean.
Checking the statements
(A) Not more than four random variables can be defined — Incorrect. Any number of functions can be defined on a sample space.
(B) Mean expectation value is a parameter and not a statistic — Correct. E(X) belongs to the distribution, not to a sample.
(C) Normal if only mean and mode are same — Incorrect. In a normal distribution the median is also equal to them.
(D) \(P(r)={}^{n}C_{r}\,q^{r}p^{n-r}\) — Incorrect. The success probability p carries the power r: \({}^{n}C_{r}\,p^{r}q^{n-r}\).
(E) Variance studies the variability of a random variable from the mean — Correct. Variance is the mean squared deviation from E(X).