probability distribution
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Probability Distribution
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Types of Random Distributions
Discrete Random Distribution Continuous Random Distribution
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Probability Distribution of Random Variable
Probability Distribution of Random Variable is defined as a table that depicts all the possible values of random variable along with their probabilities. Probability distribution of a discrete random X can be expressed as follows:
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X …….. Total
P(X) ……..
Types of Probability Distribution
Discrete Random Distribution:A random variable is said to be discrete if it takes only a finite or an infinite but countable number of values.Probability Function of a Discrete Random Distribution: If for a random variable X, the real valued function p(x) is such that
P(X=x) = p(x), Then p(x) is called probability function or probability density function of a random discrete variable. Probability function p(x) gives the measure of probability for different values of X.
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Properties of a Probability Function
If p(x) is a probability function of a random variable X, then it possesses the following properties:
1. p(x) is positive for all values of x i.e. p(x) ≥ 0 for all x.
2. ∑ p(x) = 1, summation is taken over for all values of x.
3. p(x) measures the probability for any given value of x.
4. p(x) can not be negative for any value of x. 6
Probability Mass Function
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Continuous Random Variable
A continuous random variable is a random variable that can take on any value in an interval of two values.
Height, weight, length etc. are some of the examples.
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Probability Density Function of a Continuous Random Variable
The probabilities associated with a continuous random variable X are determined by the probability Distribution function f(x) of random variable X. where 1. f(x) ≥ 0 for all values of x2. The probability that x will lie between
two numbers a and b is equal to the area under the curve y = f(x) between x=a and x=b
3. The total area under the entire curve y = f(x) is always equal to unity i.e. 1. 9
Cumulative Distribution Function of a Continuous Random Variable
Cumulative Distribution Function of a Continuous Random Variable is F(x), where
F(x) = P(X=x)=Area under the curve y= f(x) between the smallest value of X ( often -∞ ) and a point x. Properties of Cumulative Distribution
Function:1. The CDF F(X) is smooth.2. It is a non-decreasing function that
increases from 0 to 1.3. Expected value or mean is denoted by
E(X)4. The variance is denoted by V(X)
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Mathematical Expectation
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Theorems on Mathematical Expectation
Theorem 1: Expected value of a constant is constant, that is if C is constant, then
E(C) = C Theorem 2: If C is constant, then
E(CX) = C. E(X) Theorem 3: If a and b are constants,
thenE(a X ± b) = a . E(X) ± b
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Variance Variance of the probability distribution of a random variable X is the mathematical expectation of 2 . Then
Var(X) = E 2
If we put E(X) = μ then Var(X) = E 2
Var(X) = E() - For Standard Deviation () just find out the square root of the equations.
Theorems on Variance Theorem 1: If C is constant, then
V(CX) = V(X) Theorem 2: If C is constant, then
V(C) = 0 Theorem 3: If a and b are constants,
thenV(a X + b) = . V(X)
Theorem 4: If X and Y are two independent random variables, then
(i) V(X+Y)= V(X) + V(Y)(ii) V(X-Y)= V(X) + V(Y) 15
Examples1. The probability function of a random variable
X is p(x) = , x = 1, 2, 3, 4, 5, 6. Verify whether p(x) is a probability function?
2. For a random variable X, p(x) = , where x = 1, 2, 3. Is p(x) a probability density function.
3. The probability distribution of a random variable x is given below. Find (1) E(x), (2) V(x), (3) E (2x-3) and (4) V(2x-3)
(A- 0, 1.6, -3, 6.4)
4. Amit plays a game of tossing a die. If the number is less than 3 appears, he is getting Rs. A, otherwise he pays Rs. 10. If the game is fair, find a. (a=20)
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x -2 -1 0 1 2p(x) 0.2 0.1 0.3 0.3 0.1