Mathematics 3 Probability And Statistics

In a bulb factory, machines A, B and C manufactures 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A. (i) $1/3$ (ii) $2/5$ (iii) $1/5$ (iv) $3/4$

A random variable $X$ has the following probability distribution:

$X$	-2	-1	0	1	2	3
$P(X)$	0.1	$k$	0.2	$2k$	0.3	$k$

The value of $k$ is: (i) .001 (ii) .05 (iii) 0.1 (iv) 0.02

Which of the following is not a measure of Dispersion? (i) Mean Deviation (ii) Standard Deviation (iii) Quartile Deviation (iv) Average Deviation from mean.

The value of variance of a discrete random variable $X$ is given by: (i) $V(X) = E(X^2) - (E(X))^2$ (ii) $V(X) = \int x^2 f(x) dx$ (iii) $V(X) = P(X=x)$ (iv) $V(X) = \sum x^3 P(X=x)$

For a Negative skewed frequency distribution curve, the third central moment: (i) $\mu_3 > 0$ (ii) $\mu_3 < 0$ (iii) $\mu_3 = 0$ (iv) $\mu_3$ does not exist.

For a Symmetrical distribution, the coefficient of Skewness: (i) $\alpha_3 = 0$ (ii) $\alpha_3 = 3$ (iii) $\alpha_3 = 0$ (iv) $\alpha_3 = -1$

If $X$ and $Y$ are two variates, these can be at most: (i) One regression line (ii) Two regression lines (iii) Three regression lines (iv) an infinite number of regression lines

If $X$ follows a binomial distribution with parameters $n=20$ and $p=0.4$, what is the expected value $E(X)$? (i) 8 (ii) 10 (iii) 12 (iv) 6

Probability mass function for a binomial distribution with usual notations is: (i) $\binom{n}{k} p^{k-1} q$ (ii) $\binom{n}{k} p^k q$ (iii) $\binom{n}{k} p^k q^{n-k}$ (iv) $\binom{n}{k} p^k q^{n-k}$

The T-test is used to compare: (i) The means of two groups (ii) The medians of two groups (iii) The variances of two groups (iv) The standard deviations of two groups.

Two cards are drawn without replacement from a pack of 52 cards. Find the probability that: (i) both are red (ii) the first is a king and the second is an ace.

In a bulb factory, machines A, B and C manufactures 40%, 40% and 20% bulbs respectively. 4%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A.

Find the Expectation of number of heads in three tosses of fair coin. Also find (i) Mean (ii) $E(X^2)$ (iii) Variance.

The diameter $X$ of an electric cable is assumed to be continuous random variable with probability density function is: $f(x) = 6x(1-x), 0 \le x \le 1$. (i) Check that above is probability density function. (ii) Compute $P(0.5 \le x \le 0.75)$.

In a normal distribution 31% of the items are under 45 and 8% are over 64. Find the mean and variance of the distribution. [Given $f(1.4) = 0.42, f(0.5) = 0.19$].

The length of telephone conversion is an exponential variate with mean 3 minutes. Find the probability that call (a) end less than 3 min. (b) takes between 3 to 5 min.

Find the value of $k$ if $f(x, y) = k(1-x)(1-y)$ for $0 < x, y < 1$ is to be joint density function.

If the joint p.d.f. of $(X, Y)$ is $f(x, y) = 6e^{-2x-3y}, x \ge 0, y \ge 0$, find the conditional density of $Y$ given $X$.

Fit a second degree parabola to the following data:

$x$	0	1	2	3	4
$y$	1	4	10	17	30

A manufacturer claims that only 4% of his products supplied by him are defective. A random sample of 600 products contains 36 defectives. Test the claim of the manufacturer at 5% level.

marginal probability distribution

Z-Test for significance of difference of two means in case of Large samples.

Chi-square test for goodness of fit.

Large sample test for Difference of standard deviation.

In a bulb factory, machines A, B and C manufactures 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A. (i) $1/3$ (ii) $2/5$ (iii) $1/5$ (iv) $3/4$

A random variable $X$ has the following probability distribution:

$X$	-2	-1	0	1	2	3
$P(X)$	0.1	$k$	0.2	$2k$	0.3	$k$

The value of $k$ is: (i) .001 (ii) .05 (iii) 0.1 (iv) 0.02

Which of the following is not a measure of Dispersion? (i) Mean Deviation (ii) Standard Deviation (iii) Quartile Deviation (iv) Average Deviation from mean.

The value of variance of a discrete random variable $X$ is given by: (i) $V(X) = E(X^2) - (E(X))^2$ (ii) $V(X) = \int x^2 f(x) dx$ (iii) $V(X) = P(X=x)$ (iv) $V(X) = \sum x^3 P(X=x)$

For a Negative skewed frequency distribution curve, the third central moment: (i) $\mu_3 > 0$ (ii) $\mu_3 < 0$ (iii) $\mu_3 = 0$ (iv) $\mu_3$ does not exist.

For a Symmetrical distribution, the coefficient of Skewness: (i) $a_3 = 1$ (ii) $a_3 = 3$ (iii) $a_3 = 0$ (iv) $a_3 = -1$

If $X$ and $Y$ are two variates, these can be at most: (i) One regression line (ii) Two regression lines (iii) Three regression lines (iv) an infinite number of regression lines

If $X$ follows a binomial distribution with parameters $n=20$ and $p=0.4$, what is the expected value $E(X)$? (i) 8 (ii) 10 (iii) 12 (iv) 6

Probability mass function for a binomial distribution with usual notations is: (i) $\binom{n}{x} p^{k-1} q$ (ii) $\binom{n}{x} p^k q$ (iii) $\binom{n}{x} p^k q^{n-k}$ (iv) $\binom{n}{x} p^x q^n$

The T-test is used to compare: (i) The means of two groups (ii) The medians of two groups (iii) The variances of two groups (iv) The standard deviations of two groups.

Two cards are drawn without replacement from a pack of 52 cards. Find the probability that: (i) both are red (ii) the first is a king and the second is an ace.

In a bulb factory, machines A, B and C manufactures 40%, 40% and 20% bulbs respectively. 4%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A.

Find the Expectation of number of heads in three tosses of fair coin. Also find (i) Mean (ii) $E(X^2)$ (iii) Variance.

The diameter $X$ of an electric cable is assumed to be continuous random variable with probability density function is: $f(x) = 6x(1-x), 0 \le x \le 1$. (i) Check that above is probability density function. (ii) Compute $P(0.5 \le x \le 0.75)$.

In a normal distribution 31% of the items are under 45 and 8% are over 64. Find the mean and variance of the distribution. [Given $f(1.4) = 0.42, f(0.5) = 0.19$].

The length of telephone conversion is an exponential variate with mean 3 minutes. Find the probability that call (a) end less than 3 min. (b) takes between 3 to 5 min.

Find the value of $k$ if $f(x, y) = k(1-x)(1-y)$ for $0 < x, y < 1$ is to be joint density function.

If the joint p.d.f. of $(X, Y)$ is $f(x, y) = 6e^{-2x-3y}, x \ge 0, y \ge 0$, find the conditional density of $Y$ given $X$.

Fit a second degree parabola to the following data:

$x$	0	1	2	3	4
$y$	1	4	10	17	30

A manufacturer claims that only 4% of his products supplied by him are defective. A random sample of 600 products contains 36 defectives. Test the claim of the manufacturer at 5% level.

marginal probability distribution

Z-Test for significance of difference of two means in case of Large samples.

Chi-square test for goodness of fit.

Large sample test for Difference of standard deviation.

Define mutually exclusive events.

Prove that $P(\bar{A}) = 1 - P(A)$, where $\bar{A}$ is a complementary event of $A$.

If $P(A) = p_1, P(B) = p_2$ and $P(A \cap B) = p_3$ then find the value of $P(\bar{A} \cup \bar{B})$ in terms of $p_1, p_2$ and $p_3$.

An urn contains 6 white, 4 red and 9 black balls. If 3 balls are drawn at random, find the probability that one is of each colour.

A random variable $X$ follows Binomial distribution with parameters $n=40$ and $p = \frac{1}{4}$. Find the mean of $X$.

Mean of 100 observations is 50 and standard deviation is 10. What will be the new mean and standard deviation, if each observation is multiplied by 3?

Four cards are drawn at random from a pack of 52 cards. Find the probability that two are kings and two are queens.

Two unbiased dice are thrown. Find the probability that the first dice shows 6.

A random variable $X$ is such that $E[(X-1)^2] = 10$ and $E[(X-2)^2] = 6$. Find the variance of $X$.

The moment-generating function of a continuous random variable $X$ be given as $M_X(t) = (1-t)^{-9} \mid |t| < 1$. Then calculate its mean and variance.

If two dice are thrown, what is the probability that the sum is (i) greater than 8 and (ii) neither 7 nor 11?

For events $A_1, A_2, A_3, A_4, A_5, \dots, A_n$, assuming $P(\bigcup_{i=1}^n A_i) \le \sum_{i=1}^n P(A_i)$ then prove that- (i) P(\bigcap_{i=1}^n A_i) \ge \sum_{i=1}^n P(A_i) - (n-1)$; (ii) $P(\bigcap_{i=1}^n A_i) \ge 1 - \sum_{i=1}^n P(\bar{A}_i).

Find the probability that in tossing a fair coin 3 times there will appear (i) 3 heads, (ii) 2 tails and 1 head, (iii) at least 1 head and (iv) not more than 1 tail.

Ten percent of the tools produced in a certain manufacturing process turn out to be defective. Find the probability that in a sample of 10 tools chosen at random exactly two will be defective, by using (i) the Binomial distribution and (ii) Poisson approximation to the Binomial distribution.

$P(|X-3| \ge 1)$;

$P(|X-3| \ge 2)$.

0.05;

0.01.