If $z = f \left( \frac{xy}{z} \right)$ satisfies the partial differential equation $x^\alpha p - yq = \beta$, where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$, then the value of $(\alpha, \beta)$ is

The particular integral of the PDE $xy \frac{\partial^2 z}{\partial x \partial y} = 1$ is $\log(ax) \log(by)$. Then the value of $(a, b)$ is

The solution of the PDE $y^2 zp + x^2 zq = y^2 x$ is $x^\alpha - y^\beta = f(x^2 - z^2)$. Then the value of $(\alpha, \beta)$ is

Let $A$ and $B$ are two possible outcomes of an experiment and suppose $P(A) = 0.3$, $P(B) = \frac{K}{7}$ and $P(A \cup B) = 0.6$. If $A$ and $B$ are independent events, then the value of $K$ is

The first three moments of a distribution about 5 are 2, 7, 15 respectively. Then the mean and variance of the distribution are

If two dice are thrown and the probability that the sum is neither 8 nor 12 is $\frac{5}{k}$, then the value of $k$ is

A random variable $X$ has mean 2 and variance 3. If the upper bound for $P(|X-2| > 6)$ by using Chebyshev's inequality is $\frac{1}{K}$, then the value of $K$ is

If the mean of exponential distribution $f(x) = \begin{cases} ke^{-kx}, & x > 0 \\ 0, & x \le 0 \end{cases}$ is $\frac{1}{6}$, then the value of $k$ is

Let $X$ and $Y$ have joint p.d.f. $f(x, y) = \begin{cases} cxy; & 0 < x < 2, 0 < y < 1 \\ 0; & \text{elsewhere} \end{cases}$ and $E(xy) = \frac{p}{q}$. Then the value of $(p, q)$ is

Two random variables have the least square regression lines with equations $2x+y=5$ and $3x+2y=8$. Then the correlation coefficient between two variables $x$ and $y$ is

Obtain the partial differential equation by eliminating the arbitrary function $u = f(x^2 + y^2 + z^2, z^2 - 2xy)$

Solve $z^2 (p^2 + q^2) = x^2 + y^2$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$

Solve the partial differential equation $(2D^2 - DD' - 3D'^2)z = 5e^{x-y}$.

Prove that $(1-x^2)P'_n(x) = n[P_{n-1}(x) - xP_n(x)]$ where $P_n(x)$ is the Legendre's polynomial of the first kind.

Let $A$, $B$ and $C$ be three independent events. Then show that $B$ and $A \cap \overline{C}$ are also independent.

A coin is tossed. If it turns up $H$, two balls will be drawn from urn $A$, otherwise 2 balls will be drawn from urn $B$. Urn $A$ contains 3 red and 5 blue balls, urn $B$ contains 7 red and 5 blue balls. What is the probability that urn $A$ is used? It is given that both balls are blue. (Find in both cases, when balls were chosen with replacement and without replacement.)

If $X$ and $Y$ are independent Poisson variates such that $P(X=1) = P(X=2)$ and $P(Y=2) = P(Y=3)$. Find the variance of $X - 2Y$.

Show that the recurrence relation between the moments of binomial distribution is given by $\mu_{r+1} = p q \left( n r \mu_{r-1} + \frac{d\mu_r}{dp} \right)$ where $\mu_r$ is the $r$th order moment about the mean.

Compute the probability density function of $X+2Y$.

Compute the following : (i) $P(1 < y \le 2)$ (ii) $P(1 < x + 2y \le 3)$ (iii) $P(x > 1/y \le 2)$

Fit a least-square parabola of the form $y = a + bx + cx^2$ to the data.

Estimate $y$ when $x = 60$ miles per hour and 120 miles per hour. Also compute the coefficient of correlation for the above data.

If $P_n$ is the Legendre polynomial, then the value of $\int_{-1}^{1} x^m P_n dx, (m < n)$ is

If $J_n$ is the Bessel's function of first kind, then $\left[J_{-\frac{1}{2}}(x)\right]^2 + \left[J_{\frac{1}{2}}(x)\right]^2$ is equal to

The particular integral of $(D^2 - a^2 D'^2)Z = x^2$, is

The function $3x^2 + 5x - 6$ in terms of Legendre polynomial is equal to

Let the joint probability density function of the continuous random variables $X$ and $Y$ be $f(x, y) = \begin{cases} k(x^2 + y^2); & 0 < x < 1, 0 < y < 1 \\ 0; & \text{elsewhere} \end{cases}$. Then the value of $k$ is

Let $A$ and $B$ be any two events. Which one of the following is correct?

If $P(A \cap B) = \frac{1}{4}$, $P(A \cup B) = \frac{3}{4}$, $P(\overline{A}) = \frac{2}{3}$, then $P(A | B)$ is equal to

If $\mu$ is the mean and $\sigma$ is the standard deviation of a set of measurements, which are normally distributed, then the percentage of measurements within the range $\mu \pm \sigma$ is

If the density function of gamma distribution is $f(x) = \begin{cases} \frac{x^{\alpha-1} e^{-\frac{x}{\beta}}}{\beta^\alpha \Gamma\alpha}, & x > 0 \\ 0, & x \le 0 \end{cases}$. Then mean is equal to

The moment generating function of a continuous random variable $X$ be given as $M_X(t) = (1-t)^{-5}$ for $|t| < 1$. Then its mean and variance are

Solve : $(2xy-1) \frac{\partial z}{\partial x} + (z-2x^2) \frac{\partial z}{\partial y} = 2(x-yz)$

Solve : $(D^2 - DD' - 2D'^2) Z = (2x^2 + xy - y^2) \sin xy - \cos xy$

The chance that doctor $A$ will diagnose a disease $X$ correctly is 60%. The chance that a patient will die by this treatment after correct diagnosis is 40% and the chance of death by wrong diagnosis is 70%. A patient of doctor $A$, who had disease $X$, died. What is the chance that his disease was diagnosed correctly?

State axiomatic definition of the probability. Also prove that for any two events $A$ and $B$, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Fit a second degree parabola to the following data, where $x$ is the independent variable :

If $X$ and $Y$ are two uncorrelated variables and if $u = x+y$, $v = x-y$, then find the coefficient of correlation between $u$ and $v$ in terms of $\sigma_x$ and $\sigma_y$ the standard deviations of $x$ and $y$ respectively.

The first four moments of a distribution about the value 4 of the variable are -1.5, 17, -30, 108. Find first four moments about the mean.

In a distribution exactly normal, 17% of the items are under 35 and 79% items are under 63. What are the mean and standard deviation of the distribution?

If the probability that an individual suffers a bad reaction from injection of a given serum is 0.001, determine the probability that out of 1000 individuals (i) exactly 4 and (ii) more than 3 individuals will suffer a bad reaction.

The mean lifetime of a sample of 100 fluorescent light bulbs produced by a company is computed to be 1570 hours with a standard deviation of 120 hours. If $\mu$ is the mean lifetime of all the bulbs produced by the company, test the hypothesis $\mu = 1600$ hours against the alternative hypothesis $\mu < 1600$, using a level of significance of (i) 0.05 and (ii) 0.01.