A random variable X has the following probability function: (Unit 1)

i) Determine K.

ii) Evaluate $P(x < 6)$ and $P(0 \le x \le 4)$.

iii) If $P(x \le K) > (1/2)$, find the minimum value of K.

iv) Determine the distribution function of X.

v) Mean.

a) The density function of a random variable X is $f(x)=e^{-x}$ when $x \ge 0$. Find $E(x)$, $E(x^{2})$ and variance of X. (Unit 2)

b) Suppose a continuous random variable X has the probability density function $f(x)=K(1-x^{2})$ for $0 < x < 1$, and $f(x)=0$ otherwise. Find: (Unit 2)

i) K

ii) Mean

iii) Variance

a) Define Bivariate distribution. Explain their properties. (Unit 3)

b) Define Baye's theorem. The chance that doctor A will diagnose a disease X correctly is 60%. The chance that a patient will die by his 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. (Unit 3)

a) The mean and variance of a binomial variable X with parameters n and P are 16 and 8 respectively. Find $P(X \ge 1)$ and $P(X > 2)$. (Unit 4)

b) A manufacturer knows that the condensers he makes contain on average 1% defectives. He packs them in boxes of 100. What is the probability that a box picked at random will contain 3 or more faulty condensers? (Unit 4)

a) If X is a normal variate with mean 30 and standard deviation is 5. Find (Unit 4)

i) $P(26 \le X \le 40)$

ii) $P(X \ge 45)$

b) Find Karl pearson's coefficient of correlation from the following data: (Unit 4)

By the method of least squares fit a parabola of the form $y=a+bx+cx^{2}$ for the following data. (Unit 5)

a) It is claimed that a random sample of 49 tyres has mean life of 15200 km. This sampled was drawn from a population whose mean is 15150 km's and a standard deviation of 1200 km. Test the significance at 0.05 level. (Unit 5)

b) Define: (Unit 1)

i) Correlation coefficient

ii) Chebyshev's inequality

a) A sample of 26 bulbs gives a mean life of 900 hours with a standard deviation of 20 hours. The manufacturer claims that the mean life of bulbs is 1000 hours. Is the sample not up to the standard. (5% L.O.S.) (Unit 6)

b) In one sample of 10 observations, the sum of the squares of the deviations of the sample values from mean was 120 and in the other sample of 12 observations, it was 314. Test whether the difference is significant at 5% level? (Unit 6)