Probability & Statistics

Question : how to find expectation of a function of random variable ? x0123f(x)1/83/83/81/8 Find (i) E[X]        (ii) E[X^2]            (iii) E[3X-2]      (iv)  E[(X+1)^2] Solution : Fo the given probability distribution…

Question : Find expectation of the following probability distribution x0123f(x)1/83/83/81/8 Solution : Given probability distribution of a random variable X is x0123f(x)1/83/83/81/8 Since values of x are 0,1,2,3 are finite.…

Question : what are following probabilities if we toss three fair coins ? (i) Probability of getting two tails. (ii) Probability of getting two heads. (iii) Probability of getting two…

Question : Find the probability of getting two and one tail if we the toss three fair coins. Solution : Let H denotes head and T denotes tail. If we…

Question : Find the probability mass function of the heads in the tossing of three fair coins. Solution : Sample space of tossing of three fair coins is given as…

Question : Find Mean and Variance of uniform probability distribution \[f(x)\ =\frac{1}{k} \] for k= 1,2,3,….,k Solution : Given uniform probability distribution \[f\left(x_i\right)\ =\frac{1}{k} \] for i= 1,2,3,….,k This is…

Probability Function : Probability function: A function defined as \[f\left(x\right)=P\left(X=x\right)\] is called probability function of a random variable if it satisfies the following properties (i) \[f\left(x\right)>0\] (ii) \[\sum f\left(x\right)=1\] Example…

Problem 1 : For what value of k the following function works as probability function? x123456f(x)k3k5k7k9k11k Solution: A function is a probability function if it satisfies the following properties f(x)≥0…

Question : Give an example of Baye theorem of probability. Answer : Baye’s Theorem of probability : If P(A) and P(B/A),P(B/A^c) are known, then \[P\left(\frac{A}{B}\right)\ = \frac{P\left(\frac{B}{A}\right)P(A)}{P\left(\frac{B}{A}\right)P\left(A\right)+P\left(\frac{B}{A^c}\right)P(A^c)}\] Example of Baye…

Question : Given an conditional probability example. Answer : Conditional Event : If E1 , E2 are events of a sample space S and if E2 occurs after the occurrence…