Question

A fair coin is tossed four times and let x represent the number of heads which comes out

a. Find the probability distribution corresponding to the random variable x   

b. Find the expectation and variance of the probability distribution of the random variable x  

Answer 1

A Paio coin is tossed four time ,, sample space outcome is $ = {(HHHH, HHHIHHTH,HHIT, HIHH. HTHI, HITH, HIIT, IIII JTTH, TTHI, TIHH, THIÍ, THTH, THHI,} THHH 3 wher T = H: getting head. HHHH = 181 tossing got head, 2nd time get bead, 300° time got head and 4th aime also got head. Similar way the remaing x: The number of beads comes out. getting tail p(x=0) = poobability of getting o heads Number samples contains o heads Total number of samples. - 16 Similarly PC getting 1 head) - P(X=1) exactly Number of somple that contain 1 Head Number of samples in sample space HITT, TTTH, TTHI, TH THTT? 16 4 - 4

= 2 s probability of getting exadly 2 bead pex n & HHTT, HTHI HITH, ITHH, THTH} nes:) p Total number of samples. p(x= 2) 4 siipi nos.) I 16 poobability of getting exactly 8 heads. >> P (x=3) D SHHHT, HHTH; HTHH, THHI} nes) 6.16 The probability distribution of random variable x is fabulated as below q 3 4 24 L 2 5.- p(x= Y16 4/16 4416 4/16 1/16 oo b) we need to find Expectation & variance. 4 RE(X)= I 22 P(XES): 2=0 axi 1 X 4 2x4 3 x 4 4 4x4 16 + + + 16 16 16 16 E) = 1.75 28 16

The expectation of x is 1.75 Now vadiance = varca) = E(X2) - (EX)) ? 4 E(x2) 2 x præce) Lo - OX 4 * 4 + + 9*4 1x4 16 + + 16 * 1 16 16 16 16 72 16 4.5 .: variance = E(X2) -(Ecx)) 4.5 - (175) variance CX) = 1.4375 The variance. of x 16 /1.4375
Hope this will help you. Thank you.