**Important RGPV Question **

Table of Contents

Toggle**AL-302Β Introduction to Probability and Statistics**

**III Sem, AIML**

**UNIT 1-Β **Basic Probability

**Q.1)Β **Define discrete random variable and independence of random variable. Also, show that

E(X_{1}+X_{2}+…+X_{n}) = E(X_{1}) + E(X_{2})+…+E(X_{n})

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**Q.2)Β **Define expectation of random variables. Also, show thatΒ V(aX+b)= aΒ²V(X)

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**Q.3)** State Chebyshev’s inequality. If nββ, pβ0 and np=2, then show that binomial distribution reduces to Poisson distribution.

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**Q.4)Β **Show that Poisson distribution is a limiting case of binomial distribution under the case

nββ, pβ0 and np = m.

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**Q.5)**Β Show that sum of two independent random variables follows normal distribution.

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**Q.6)Β **Define Poisson distribution and obtain its mean and variance.

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**Q.7)Β **Define binomial distribution and derive its moment generating function. Hence, obtain its mean and variance.

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**Q.8)Β **A random variable X has the following probability function:

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β

i) Determine K

ii) Evaluate P(x < 6) and P(0 β€xβ€4)

iii) If P(xβ€K) > (1/2) find the minimum value of K

iv) Determine the distribution function of X

v) Mean

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**Q.9)Β **The density function of a random variable X is f(x)=e^{-x}Β when x β₯0.. Find E(x), E(xΒ²) and variance of X.

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**Q.10)Β **Suppose a continuous random variable X has the probability density function/(x)=K(1-x^{2}) for 0<x<1, and f(x)=0 otherwise. Find:

i) K

ii) Mean

iii) Variance

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**Q.11) **Define-

i) Correlation coefficient

ii) Chebyshev’s inequality

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**Β UNIT 2-Β **Continuous Probability Distributions

**Q.1)Β **Define gamma distribution. If X follows exponential distribution with parameter 6, then obtain its mean and variance.

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**Q.2)Β **Define normal distribution with its properties. Describe the methodology for difference of means for large samples.

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**Q.3)** Define exponential distribution with parameter Ξ and obtain its mean, variance and moment generating function.

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**Q.4)Β **Define continuous random variable. Also show that

E(X_{1}X_{2…}X_{n}) = E(X_{1}) E(X_{2})+…+E(X_{n})

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**Q.5)** Define gamma distribution. If X follows exponential distribution with parameter 6, then obtain its mean and variance.

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**Q.6)Β **Define gamma distribution with parameter 2 and obtain its mean, variance, and characteristic function.

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**UNIT 3-Β **Bivariate Distributions

**Q.1)Β **Describe bivariate distribution. Write down the probability density function of bivariate normal distribution.

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**Q.2)Β **Define Bivariate distribution. Explain their properties.

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**Q.3)**Β 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.

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**UNIT 4-Β **Basic Statistics

**Q.1)Β **Define regression coefficient with its properties. If

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**Q.2)Β **What do you understand by measure of central tendency? Also, write down its merits and demerits. Calculate mean and standard deviation for the observations 5, 10, 20, 25, 40, 42, 45, 48, 70, 80.

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**Q.3)** Define Binomial distribution and obtain its mean and variance.

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**Q.4)Β **What do you mean by measure of skewness? Write tests of skewness.

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**Q.5)** Define Spearman’s rank correlation coefficient and obtain rank correlation coefficient for the following data

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**Q.6)Β **Find the coefficient of correlation between X and Y

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**Q.7)Β **Find the regression line of y on x for the following data:Β Β Β

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**Q.8)Β **Define non-central and central moments. Also, show that

V(2X+3)=4V(X)

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**Q.9)Β **If XβΌN(ΞΌ,ΟΒ²), then show thatΒ

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**Q.10)Β **Write down merits and demerits of measure of central tendency. Find the median for the following data;Β Β Β Β Β

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**Q.11) **What do you mean by measure of kurtosis? Obtain Bi and B, for the following function:Β f(x)=y_{0}x(2-x); 0β€xβ€2

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**Q.12) **Find the rank correlation coefficient to the following data:

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**UNIT 5-Β **: Applied Statistics

**Q.1)Β **Fit a second degree parabola to the following data:

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**Q.2) **Using method of Least squares, find the curve y = ax + arΒ² that best fit the following data:Β Β Β Β **(RGPV Jan 2023)**

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β

**Q.3)**Β ) If X is a normal variate with mean 30 and standard deviation is 5. Find 7

i) P(26β€ X β€40)

ii) P(Xβ₯45)

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**Q.4)Β **By the method of least squares fit a parabola of the form y=a+bx+cxΒ² for the following data.Β Β Β

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**Q.5)**Β 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.

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**Q.6)**Β 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.)

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**Q.7)Β **In one sample of 10 observations, the sum pf 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?

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**UNIT 6-Β **Small samples

**Q.1)**Β What do you understand by Chi-square test of goodness of fit? Write condition for applying Chi-square test.

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**EXTRA QUESTIONS-**

**Q.1)Β **A large number of measurement is normally distributed with a mean 65.5 cm and S.D. of 6.2 cm. Find the percentage of measurement that fall between 54.8 cm and 68.8 cm.

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**Q.2)Β **Out of 8000 graduates in a town, 800 are females, out of 1600 graduate employees, 120 are females, Use xΒ² test to determine if any distinction is made in appointment on the basis of sex. The value of xΒ² for 1 degree of freedom at 5% level is 3.841.

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**Q.3)**Β Define Karl Pearson’s correlation coefficient and obtain the correlation coefficient for the following data:Β

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**Q.4)Β **The mean and variance of a binomial variable X with parameters n and P are 16 and 8 respectively. Find P(Xβ₯ 1) and P(X > 2).

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**Q.5)** 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?

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**Q.6)Β **Find Karl pearson’s coefficient of correlation from the following data

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**— Best of Luck for Exam —**

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