1. Define probability of an event.

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Probability of an event is defined as - Probability of an event happening= No of ways it can happen/total no of outcomes.

2.Define conditional probability and Baye’s rule.

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Conditional probability is the probability of an event occurring given that another event has already occurred. The baye’s theorem or baye’s rule describes the probability if an event, based on prior knowledge of the conditions that might be related to the event.

3.What do you understand by statistically independent events?

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The two events A and B are statistically independent if and only if
  P (AnB) = P (A).P (B).
Similarly X and Y are statistically independent random variables if and only if P {X≤ x, Y≤ y} = P {X≤x} P {Y≤ y}
  Fxy(y)=Fx(x).Fy(y)
  Fx(y)=fx(x)fy(y)

4. Define the term random variable.

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Random variable is a variable whose value is unknown or a function that assigns value to each of an experiment’s outcome.

5.Explain the meaning of the term discrete RV and continuous RV.

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A discrete random variable X has a countable number of possible values. Whereas, a continuous random variable X takes all values in a given interval of numbers.

6.State important properties of CDF.

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1. Every CDF Fx is non decreasing and right continuous
lim_x→-∞F_x(x) = 0 and lim_x→+∞F_x(x) = 1
2. For all real numbers a and b with continuous random variable X, then the function fx is equal to the derivative of Fx, such that

3. If X is a completely discrete random variable, then it takes the values x1, x2, x3,… with probability pi = p(xi), and the CDF of X will be discontinuous at the points xi:

7. Define PDF

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The derivative of cumulative distributive function (CDF) with respect to some dummy variable is known as Probability Density Function (PDF).

8. Define joint probability density function for two random variables.

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Given 2 random variables X and Y, a joint probability density function or joint pdf is the density of probability for joint events i.e. if F is a subset of the real plane, then
  P(F) = P(X, Y) € F =⌡⌡fx, y(y)

9. Define Central Limit Theorem.

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The central limit theorem states that the random variable X which is the sum of the large number of random variables always approaches the Gaussian distribution irrespective of the type of distribution each variable process and their amount of contribution into the sum. X3=X1+X2

10.Define uniform distribution function?

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The uniform distribution function has a random variable X restricted to a finite interval {a, b} and has f(x) as constant density over the interval. The function f(x) is defined by: F(x) = {1/ (b-a), a≤x≤b 0, otherwise}