Consider the following statements with respect to probability distributions:
A. When mean \((\mu) = 1\) and standard deviation \((\sigma) = 0\) for a data set, normal distribution is called standard normal distribution.
B. In a normal distribution of data, z is given by \(z = \displaystyle\frac{\mu - x}{\sigma}\)
C. P(‘r’ success) is the \((r+1)^{\text{th}}\) term in the binomial expansion of \((q+p)^{n}\).
D. In a random experiment, a collection of trials is called Bernoulli, if trials are dependent by nature.
E. When a random variable whose value is obtained by measuring and it takes many values between two values, it is called a continuous random variable.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → C and E only
The binomial probability of r successes in n trials is
\(P(X = r) = {}^{n}C_{r}\, q^{n-r} p^{r}\)
and the terms of \((q+p)^{n}\) are \({}^{n}C_{0}q^{n}, {}^{n}C_{1}q^{n-1}p, \ldots\), so the term with \(p^{r}\) is the \((r+1)^{\text{th}}\) term.
So the standard normal distribution has \(\mu = 0\), \(\sigma = 1\); the standard normal variate is \(z = \frac{x - \mu}{\sigma}\); Bernoulli trials are independent; and P(r successes) is the \((r+1)^{\text{th}}\) term of \((q+p)^{n}\).
Checking the statements
(A) When mean \((\mu) = 1\) and standard deviation \((\sigma) = 0\) for a data set, normal distribution is called standard normal distribution. — Incorrect. The standard normal distribution has mean 0 and standard deviation 1.
(B) In a normal distribution of data, z is given by \(z = \frac{\mu - x}{\sigma}\) — Incorrect. The numerator is reversed; \(z = \frac{x - \mu}{\sigma}\).
(C) P(‘r’ success) is the \((r+1)^{\text{th}}\) term in the binomial expansion of \((q+p)^{n}\). — Correct. \({}^{n}C_{r}q^{n-r}p^{r}\) is the \((r+1)^{\text{th}}\) term, since the first term has \(r = 0\).
(D) In a random experiment, a collection of trials is called Bernoulli, if trials are dependent by nature. — Incorrect. Bernoulli trials are independent, with the same probability of success in each.
(E) When a random variable whose value is obtained by measuring and it takes many values between two values, it is called a continuous random variable. — Correct. A measured variable that can take any value in an interval is continuous.